EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 43-52 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bi-interior ideal elements in ∧e-semigroups Niovi Kehayopulu Abstract. All the results on semigroups obtained using only sets, can be written in an abstract form in a more general setting. Let us consider a recent paper to justify what we say. The bi- interior ideals of semigroups introduced and studied by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. in 2018, follow for more general statements about ordered semigroups. The same holds for every result of this sort on semigroups based on right (left) ideals, bi-ideals, quasi- ideals, interior ideals etc. for which we use sets. As a result, we have an abstract formulation of the results on semigroups obtained by sets that is in the same spirit with the abstract formulation of general topology (the so-called topology without points) initiated by Koutský, Nöbeling and, even earlier, by Chittenden, Terasaka, Nakamura, Monteiro and Ribeiro. As a consequence, results on ordered Γ-hypersemigroups and on similar simpler structures can be obtained. 2020 Mathematics Subject Classifications: 06F05, 20M10 Key Words and Phrases: ∧e-semigroup, right (left), bi-ideal, quasi-ideal element, bi-interior ideal element, left simple, simple, bi-interior simple, regular 1. Introduction and prerequisites The concept of bi-interior ideal of semigroup has been introduced my M.M. Krishna Rao [4] as follows: Let S be a semigroup and A a nonempty subset of S. Then A is called a bi-interior ideal of S if ASA ∩ SAS ⊆ A. As one can easily see, every bi-ideal A of S is a bi-interior ideal of S and every interior ideal of S is a bi-interior ideal of S. So the concept of bi-interior ideal generalizes the concept of bi-ideal and the concept of interior ideal. As every right (resp. left) ideal and every quasi-ideal of a semigroup S is a bi-ideal of S, the concept of bi-interior ideal generalizes the concepts of right ideal, left ideal, and the concept of quasi-ideal of a semigroup as well. M. Murali Krishna Rao assumes that the bi-ideals and the interior ideals of a semigroup S are subsemigroups of S but this does not make any difference to the investigation. The results of [4] follows from a more general setting of that of ordered ∧e-semigroups. The same can be said for any similar result based on sets. We casually chose a recent paper by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. in 2018 as an example to justify what we say. This is in the same spirit with the abstract formulation of general DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.3931 Email address: nkehayop@math.uoa.gr http://www.ejpam.com 43 © 2021 EJPAM All rights reserved. N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 44 topology (the so-called topology without points) initiated by Koutský and Nóbeling [3, 5]. Topology without points has been also studied much earlier by M. Nakamura [Closure in general lattices. Proc. Imp. Acad. Tokyo 17, 5–6 (1941); MR0004225], A. Monteiro and H. Ribeiro [L’operation de fermeture et ses invariants dans les systèmes partiellement ordonnées. Portugal. Math. 3 (1942), 171–184; MR0007973] or, even earlier, by E.W. Chittenden [On general topology and the relation of the properties of the class of all continuous functions to the properties of space. Trans. Amer. Math. Soc. 31, no. 2 (1929), 290–321; MR1501484] and H. Terasaka [Die Theorie der topologischen Verbände. Coll. Papers Fac. Sci. Osaka Univ. Ser. A 8, no. 1 (1940), 33 pp.; MR0032581]. The following definitions are well known: If S is a semigroup, a nonempty subset A of S is called a right (resp. left) ideal of S if AS ⊆ A (resp. SA ⊆ A). It is called a bi-ideal of S is ASA ⊆ A (Kehayopulu); and quasi-ideal of S if AS ∩SA ⊆ A. S. Lajos considered the bi-ideal of a semigroup S as a subsemigroup of S except in his last publications in which he used the definition given above. It might be mentioned that most of the results hold without the assumption that the bi-ideal is a subsemigroup; as so we do not have to use the term “generalized bi- ideal” so often. A poe-groupoid is a groupoid S at the same time an ordered set having a greatest element “e” (: e ≥ a for every a ∈ S) such that a ≤ b implies ac ≤ bc and ca ≤ cb for every c ∈ S. If the multiplication on S is associative, then S is called poe-semigroup. An le-semigroup is a semigroup S at the same time a lattice having a greatest element e (with respect to the order) such that a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc for every a, b, c ∈ S. Every le-semigroup is a poe-semigroup. A ∧e-groupoid is a groupoid S at the same time a semilattice under ∧ (: ∧-semilattice) having a greatest element “e” such that a ≤ b implies ac ≤ bc and ca ≤ cb for every c ∈ S; if its multiplication is associative, then it is called ∧e-semigroup. Let S be a poe-groupoid. An element a of S is called right (left) ideal element of S if ae ≤ a (resp. ea ≤ a). An element that is both a right and a left ideal element is called ideal element. If S is a ∧e-groupoid, then an element a of S called a quasi-ideal element if ae ∧ ea ≤ a. An element a of a poe-semigroup is called a bi-ideal element if aea ≤ a and an interior ideal element if eae ≤ a. Denote by r(a) (resp. l(a)) the right (resp. left) ideal of S generated by a. For an le-semigroup S, we have r(a) = a ∨ ae and l(a) = a ∨ ea. An element a of a poe-groupoid is called subidempotent if a2 ≤ a; it is called idempotent if a2 = a [1]. An element e′ of a poe-groupoid S is called an identity (or unity) of S is ae′ = e′a = a for every a ∈ S. The study of poe-semigroups plays an essential role in the theory of ordered Γ-hyper- semigroups and related simpler structures, like the hypersemigroups, for example. 2. Bi-interior ideal elements in ∧e-semigroups Proposition 2.1. If S is a ∧e-groupoid, then every right (resp. left) ideal element of S is a quasi-ideal element of S. If S is a ∧e-semigroup, then every quasi-ideal element of S is a bi-ideal element of S. Proof. Let a be a right ideal element of S. Then ae∧ea ≤ ae ≤ a and so a is a quasi-ideal N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 45 element of S. If a is a quasi-ideal element of S, then aea ≤ ae ∧ ea ≤ a and so a is a bi-ideal element of S. □ Definition 2.2. An element b of a ∧e-semigroup S is called a bi-interior ideal element if beb ∧ ebe ≤ b. Proposition 2.3. Let S be a ∧e-semigroup. Then we have the following: (1) Every right (resp. left) ideal element of S is a bi-interior ideal element of S. (2) Every quasi-ideal element of S is a bi-interior ideal element of S. (3) Every bi-ideal element of S is a bi-interior ideal element of S. (4) Every interior ideal element of S is a bi-interior ideal element of S. Proof. (3) If b be a bi-ideal element of S, then beb ∧ ebe ≤ beb ≤ b, so b is a bi-interior ideal element of S. (4) If b is an interior ideal element of S, then beb ∧ ebe ≤ ebe ≤ b, so b is a bi-interior ideal element of S. (2) If q is a quasi-ideal element of S then, by Proposition 2.1, q is a bi-ideal element of S so, by (3), q is a bi-interior ideal element of S. Independently, if q is a quasi-ideal element of S, then qeq ∧ eqe ≤ qeq ≤ qe ∧ eq ≤ q, so q is a bi-interior ideal element of S. (1) If a is a right (resp. left) ideal element of S then, by Proposition 2.1, a is a quasi-ideal element of S so, by (2), a is a bi-interior ideal element of S. □ Proposition 2.4. Let S be a ∧e-semigroup. Then we have the following: (1) If a and b are bi-interior ideal elements of S, then a∧ b is a bi-interior ideal element of S. (2) If a is a right ideal element and b is a left ideal element of S, then a∧b is a bi-interior ideal element of S. (3) If b is a bi-interior ideal element and t is an interior ideal element of S, then b ∧ t is a bi-interior ideal element of S. Proof. (1) Let a and b be bi-interior ideal elements of S. Then aea∧eae ≤ a and beb∧ebe ≤ b, then (a ∧ b)e(a ∧ b) ∧ e(a ∧ b)e ≤ aea ∧ eae ≤ a and (a ∧ b)e(a ∧ b) ∧ e(a ∧ b)e ≤ beb ∧ ebe ≤ b. Thus we have (a ∧ b)e(a ∧ b) ∧ e(a ∧ b)e ≤ a ∧ b N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 46 and so a ∧ b is a bi-interior ideal element of S. (2) If a is a right ideal element of S and b is a left ideal element of S then, by Proposition 2.3(1), a and b are bi-interior ideal elements of S and then, by property (1), a ∧ b is a bi-interior ideal element of S. (3) If b is a bi-interior ideal element and t is an interior ideal element of S then, by Proposition 2.3(4), a and b are bi-interior ideal elements of S so, by (1), b∧t is a bi-interior ideal element of S. □ Proposition 2.5. Let S be a ∧e-semigroup. Then we have the following: (1) If b is a bi-interior ideal element of S, then the elements be and eb are bi-interior ideal elements of S as well. (2) If b is a bi-interior ideal element of S, b ≤ be or b ≤ eb, then b is a subidempotent element of S. (3) If b is a bi-interior ideal element of S and (the greatest element) e is at the same time the identity of S, then b is a subidempotent element of S. Proof. (1) Let b be a bi-interior ideal element of S. Then (be)e(be) ∧ e(be)e ≤ (be)e(be) ≤ be and (eb)e(eb) ∧ e(eb)e ≤ (eb)e(eb) ≤ eb, so be and eb are bi-interior ideal elements of S. (2) Let b be a bi-interior ideal element of S such that b ≤ be. Then we have b2 ≤ (be)b and b2 ≤ b(be) ≤ ebe, thus we have b2 ≤ beb∧ebe ≤ b and so b is subidempotent. If b ≤ eb, then b2 ≤ b(eb) and b2 ≤ (eb)b ≤ ebe, then b2 ≤ beb ∧ ebe ≤ b and so b is subidempotent. (3) Since b = be = eb, the proof follows from (2). □ Propositions 2.3, 2.4 and 2.5 generalize the Theorem 3.3 in [4]. Proposition 2.6. Let S be a ∧e-semigroup. Then we have the following: (1) If a is a right ideal element of S then, for any b ∈ S, the element ab is a bi-interior ideal element of S. (2) If a is a left ideal element of S then, for any b ∈ S, the element ba is a bi-interior ideal element of S. (3) For any a, b ∈ S, the element aeb is a bi-interior ideal element of S. N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 47 Proof. (1) Let a be a right ideal element of S and b ∈ S. Then (ab)e(ab) ∧ e(ab)e ≤ (ab)e(ab) ≤ (ae)b ≤ ab. (2) Let a be a left ideal element of S and b ∈ S. Then (ba)e(ba) ∧ e(ba)e ≤ (ba)e(ba) ≤ b(ea) ≤ ba. (3) Let a, b ∈ S. The element aeb is a bi-ideal element of S. In fact, (aeb)e(aeb) ≤ aeb. Then, by Proposition 2.3(3), aeb is a bi-interior ideal element of S. □ Proposition 2.7. The following assertions are satisfied: (1) If S is a ∧e-semigroup and b, t ∈ S such that tet∧ ete ≤ b ≤ t, then b is a bi-interior ideal element of S. (2) If S is an ∧e-semigroup and semilattice under ∨ at the same time and b, t ∈ S such that tet ∨ ete ≤ b ≤ t, then b is a bi-interior ideal element of S. Proof. (1) We have beb ∧ ebe ≤ tet ∧ ete ≤ b, so b is a bi-interior ideal element of S. (2) We have beb ≤ tet ≤ tet ∨ ete ≤ b and ebe ≤ ete ≤ tet ∨ ete ≤ b; thus we have beb ∧ ebe ≤ b and so b is a bi-interior ideal element of S. □ Proposition 2.8. Let S be a ∧e-semigroup. If b is a bi-interior ideal element of S and t ∈ S such that t ≤ b ≤ bt, then bt is a bi-interior ideal element of S. Proof. Let b be a bi-interior ideal element of S and t ≤ b ≤ bt. Then we have (bt)e(bt) ∧ e(bt)e ≤ bet ∧ ebe ≤ beb ∧ ebe ≤ b ≤ bt, thus bt is a bi-interior ideal element of S. □ Theorem 2.9. Let S be a ∧e-semigroup such that x ≤ xe for every x ∈ S. Let a be a minimal right ideal element and b a minimal left ideal element of S. Then ab is a minimal bi-interior ideal element of S. Proof. Since a is a right ideal element of S, we have (ab)e(ab) ≤ (ae)b ≤ ab, then ab is a bi-ideal element of S and so ab is a bi-interior ideal element of S (by Prop. 2.3(3)). Let now z be a bi-interior ideal element of S such that z ≤ ab. Then ez ≤ e(ab) ≤ eb ≤ b and ze ≤ (ab)e ≤ ae ≤ a. Since ez is a left ideal element of S, ez ≤ b, and b is a minimal left ideal element of S, we have ez = b. Since ze is a right ideal element of S, ze ≤ a, and a is a minimal right ideal element of S, we have ze = a. Then we have ab = (ze)(ez) ≤ zez. By hypothesis, we have ab ≤ (ab)e = abe = (ze)(ez)e ≤ eze. Thus we have ab ≤ zez∧eze ≤ z. Then we obtain z = ab and the proof is complete. □. Corollary 2.10. (cf. also [4; Theorem 3.10]) Let S be a semigroup such that A ⊆ AS for every A ⊆ S. If A is a minimal right ideal and B is a minimal left ideal of S, then the product AB is a minimal bi-interior ideal of S. N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 48 3. Bi-interior ideal elements in left simple, simple and bi-interior simple ∧e-semigroups Definition 3.1. A poe-groupoid S is said to be left (resp. right) simple if for every left (resp. right) ideal element a of S we have a = e. That is, if e is the only left (resp. right) ideal element of S. It is called simple if for every ideal element a of S we have a = e; that is, if e is the only ideal element of S. If S is left (or right) simple, then it is simple. Proposition 3.2. If S is a left (resp. right) simple ∧e-semigroup and b is a bi-interior ideal element of S, then b is a right (resp. left) ideal element of S. Proof. Let S be left simple and b a bi-interior ideal element of S. Since eb is a left ideal element of S and S is left simple, we have eb = e. Since beb ∧ ebe ≤ b, we get be ∧ e2 ≤ b. Since e2 is a left ideal element of S and S is left simple, we have e2 = e. Then we have be = be ∧ e ≤ b, then be ≤ b and so b is a right ideal element of S. □ Since every bi-ideal element of S is a bi-interior ideal element of S (Prop. 2.3(3)), by Proposition 3.2 we have the following. Corollary 3.3. If S is a left (resp. right) simple ∧e-semigroup and b is a bi-ideal element of S, then b is a right (resp. left) ideal element of S. Proposition 3.4. If S is a simple ∧e-semigroup, then every bi-interior ideal element of S is a bi-ideal element of S. Proof. Let b be a bi-interior ideal element of S. Then beb∧ ebe ≤ b. Since ebe is an ideal element of S and S is simple, we have ebe = e. Then beb ∧ e ≤ b and so beb ≤ b. □ Following M. Murali Krishna Rao, we give the following definition. Definition 3.5. A ∧e-semigroup S is said to be bi-interior simple if, for every bi-interior ideal element b of S, we have b = e. Proposition 3.6. Let S be a ∧e-semigroup. Then S is bi-interior simple if and only if, for every a ∈ S, we have aea ∧ eae = e. Proof. =⇒. Let a ∈ S. The element aea∧ eae is a bi-interior ideal element of S. Indeed, (aea ∧ eae)e(aea ∧ eae) ∧ e(aea ∧ eae)e ≤ aeaeaea ∧ eaeae ≤ aea ∧ eae. Since S is bi-interior simple, we have aea ∧ eae = e. ⇐=. Let b be a bi-interior ideal element of S. By hypothesis, we have e = beb ∧ ebe ≤ b. Then b = e and so S is bi-interior simple. □ Corollary 3.7. [4; Theorem 3.5] A semigroup M is bi-interior simple if and only if MaM ∩ aMa = M for every a ∈ M . N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 49 4. Bi-interior ideal elements in regular ∧e-semigroups A poe-semigroup S is said to be regular if, for every x ∈ S, we have x ≤ xex [1]. Theorem 4.1. Let S be a ∧e-semigroup. If S is regular then, for every bi-interior ideal element b of S, we have beb ∧ ebe = b. In particular, if S is an le-semigroup, then S is regular if and only if for every bi-interior ideal element b of S, we have beb ∧ ebe = b. Proof. =⇒. Let b be a bi-interior ideal element of S. Then beb ∧ ebe ≤ b. Since S is regular, we have b ≤ beb ≤ (beb)e(beb) ≤ beb ∧ ebe. Thus beb ∧ ebe = b. ⇐=. Let a be a right ideal element and and b a left ideal element of S. By Proposition 2.4(2), a∧b is a bi-interior element of S. By hypothesis, we have (a∧b)e(a∧b)∧e(a∧b)e = a ∧ b. Then we have a ∧ b ≤ (a ∧ b)e(a ∧ b) ≤ (ae)b ≤ ab ≤ ae ∧ eb ≤ a ∧ b. Then a ∧ b = ab. Thus, for any x ∈ S, we have x ≤ r(x) ∧ l(x) = r(x)l(x) = (x ∨ xe)(x ∨ ex) = x2 ∨ xex, then x2 ≤ x3 ∨ xex2 ≤ xex, then x ≤ xex and so S is regular. □ Theorem 4.2. Let S be a ∧e-semigroup. If S is regular, then every bi-interior ideal element of S is subidempotent. “Conversely”, if S is an le-semigroup, then S is regular if and only if every bi-interior ideal element of S is idempotent. Proof. =⇒. Let b be a bi-interior ideal element of S. Since S is regular, we have b ≤ beb. Then we have b2 ≤ (beb)b ≤ beb ∧ ebe = b, thus b is subidempotent. ⇐=. Let a be a right ideal element and b a left ideal element of S. By Proposition 2.4(2), a ∧ b is a bi-interior ideal element of S. By hypothesis, we have a ∧ b = (a ∧ b)2 = (a ∧ b)(a ∧ b) ≤ ab ≤ ae ∧ eb ≤ a ∧ b, thus a ∧ b = ab, and S is regular (see the proof of Theorem 4.1). □ Proposition 4.3. Let S be a regular ∧e-semigroup. Then b is a bi-interior ideal element of S if and only if b is a bi-ideal element of S. Proof. =⇒. Let b be a bi-interior element of S. Since S is regular, we have b ≤ beb ≤ (beb)e(beb) ≤ (beb) ∧ (ebe) ≤ b. Thus we have b = beb, and b is a bi-ideal element of S. The ⇐-part follows from Proposition 2.3(3), and it holds for ∧e-semigroups in general. □ Theorem 4.4. Let S be a regular ∧e-semigroup. Then b is a bi-interior ideal element of S if and only if there exists a right ideal element r and a left ideal element l of S such that b = rl. N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 50 Proof. =⇒. Let b be a bi-interior ideal element of S. Since S is regular, by Theorem 4.1, we have beb ∧ ebe = b. The element be and eb are right and left ideal elements of S, respectively. It is enough to prove that b = (be)(eb). Since S is regular, we have b ≤ beb. Thus we have (be)(eb) ≤ (beb)e(beb) ≤ (beb) ∧ (ebe) = b. We also have b ≤ beb ≤ (beb)e(beb) ≤ (be)(eb) and so b = (be)(eb). The “⇐-part follows by Proposition 2.6(1) (or 2.6(2)) and it holds for ∧e-semigroups in general. □ Corollary 4.5. (cf. also [4; Theorem 3.28]) Let M be a regular semigroup. Then B is a bi-interior ideal of M if and only if there exists a right ideal R and a left ideal L of M such that B = RL. Proposition 4.6. Let S be a poe-semigroup, b a subidempotent bi-ideal element of S and a ∈ S such that a ≤ b and a = aba. Then a is a bi-interior ideal element of S. Proof. Since a ≤ b, we have ba ≤ b2 ≤ b and ab ≤ b2 ≤ b. Then a = a(ba) ≤ ab and a = (ab)a ≤ ba. Then aea ≤ (ab)e(ba) = a(beb)a ≤ aba = a and so a is a bi-ideal element of S. Then, by Proposition 2.3(3), it is a bi-interior ideal element of S as well. □ Theorem 4.7. A ∧e-semigroup S is regular if and only if for every bi-interior ideal element b, every ideal element i and every left ideal element l of S, we have b∧ i∧ l ≤ bil. Proof. =⇒. Let b be a bi-interior ideal element, i an ideal element and l a left ideal element of S. Since S is regular, we have b ∧ i ∧ l ≤ (b ∧ i ∧ l)e(b ∧ i ∧ l) ≤ ( (b ∧ i ∧ l)e(b ∧ i ∧ l)e(b ∧ i ∧ l) )( e(b ∧ i ∧ l)e(b ∧ i ∧ l) ) e(b ∧ i ∧ l). We have b ∧ i ∧ l ≤ b, e(b ∧ i ∧ l)e ≤ e, (b ∧ i ∧ l) ≤ b and so (b ∧ i ∧ l)e(b ∧ i ∧ l)e(b ∧ i ∧ l) ≤ beb. (b ∧ i ∧ l)e ≤ e, b ∧ i ∧ l ≤ b, e(b ∧ i ∧ l) ≤ e and so (b ∧ i ∧ l)e(b ∧ i ∧ l)e(b ∧ i ∧ l) ≤ ebe. Thus we have (b ∧ i ∧ l)e(b ∧ i ∧ l)e(b ∧ i ∧ l) ≤ beb ∧ ebe ≤ b. Moreover, e(b ∧ i ∧ l)e(b ∧ i ∧ l) ≤ eiei ≤ i and e(b ∧ i ∧ l) ≤ el ≤ l. Hence we obtain b ∧ i ∧ l ≤ bil. ⇐=. Let a be a right ideal element and b a left ideal element of S. Since a is a bi-interior ideal element, e an ideal element and b a left ideal element of S, by hypothesis, we have a ∧ b = a ∧ e ∧ b ≤ aeb ≤ ab ≤ ae ∧ eb ≤ ab, then a ∧ b = ab and so S is regular. □ Example 4.8. We consider the ∧e-semigroup S = {a, b, c, d, e} given by Table 1 and Figure 1. This is an le-semigroup at the same time. N. Kehayopulu / Eur. J. Pure Appl. Math, 14 (4) (2021), 43-52 51 · a b c d e a e b a d e b b b b b b c a b c d e d d b d d d e e b e d e Table 1 b cd e a Figure 1 This is regular as x ≤ xex for every x ∈ S. The bi-interior ideal elements of S are the sets b, d and e. The results of sections 2 and 4 can be applied. This is not left simple, right simple, simple or bi-interior simple. Note. We do not have to assume that all semigroups in [4] have unity. In case we need it, the assumption A ⊆ AM and A ⊆ MA for every nonempty subset A of S provides a more general situation. It is not known if the Theorem 3.33 in [4] holds since its proof is wrong. The proof of the “⇒”-part of Theorem 3.14 in [4] is wrong; however the above Theorem 4.7 shows that it can be proved and the Theorem 3.14 in [4] holds. 5. Conclusion The results of the present paper generalize corresponding results by M.M. Krishna Rao in Discuss. Math. Gen. Algebra Appl. In a similar way all the results on semigroups based on sets can be written in an abstract form using elements (instead of sets). REFERENCES 52 References [1] N. Kehayopulu. On intra-regular ∨e-semigroups. Semigroup Forum 19(2):111–121, 1980. [2] N. Kehayopulu. Note on interior ideals, ideal elements in ordered semigroups. Sci. Math. 2(3):407–409, 1999. [3] K. Koutský. Théory the lattices topologiques. Publ. Fac. Sci. Univ. Masaryk 95:133– 171, 1952. [4] M.M. Krishna Rao. Bi-interior ideals of semigroups. Discuss. Math. Gen. Algebra Appl. 38(1):69–78, 2018. [5] G. Nöbeling, Grundlagen der analytischen Topologie. Die Grundlehren der mathema- tischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung der Anwendungsgebiete, Bd. LXXII. Springer-Verlag, Berlin-Gottingen-Heidelberg, 1954. x+221 pp.