() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 1-13 The structure of the poset of regular topologies on a set Ofelia T. Alas and Richard G. Wilson Abstract We study the subposet Σ3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R-closed does have such a predecessor and as a corollary we obtain a result of Costantini that each non-compact Tychonoff space has an immediate predecessor in Σ3. We also consider the problem of when an R-closed topology is maximal R-closed. 2010 MSC: Primary 54A10; Secondary 06A06, 54D10 Keywords: Lattice of T1-topologies, poset of T3-topologies, upper topology, lower topology, R-closed space, R-minimal space, submaximal space, maximal R-closed space, dispersed space 1. Introduction In a previous paper [3], we studied the problem of when a jump can occur in the order of the lattice L1(X); that is to say, when there exist T1-topologies τ and τ+ on a set X such that whenever µ is a topology on X such that τ ⊆ µ ⊆ τ+ then µ = τ or µ = τ+. The existence of jumps in L1(X) and in the subposet of Hausdorff topologies, has been studied in [5], [2], [10] and [16]; in the last two articles an immediate successor τ+ was said to be a cover of (or simply to cover) τ. In the above cited paper [3], when a topology τ has a cover τ+ we have called τ a lower topology and τ+ an upper topology and we continue to use this terminology here. In the present work we study the structure of the subposet Σ3(X) of all T3-topologies of the lattice L1(X), on a set X with a view to deciding which elements of this partially ordered sets have and which do not have covers. 2 O. T. Alas and R. G. Wilson In [1] it was shown that a T3-topology on X which is not feebly compact is an upper topology in Σ3(X) and in [6], Costantini showed that every non- compact Tychonoff topology on X is upper in Σ3(X). In Section 2 of this paper we generalize both these results by showing that every T3-topology which is not R-closed is upper in Σ3(X). (A T3-space is R-closed if it is closed in every embedding in a T3-space.) In Section 3 we consider the problem of the existence of spaces which are maximal with respect to being R-closed and in Section 4 we study lower topologies in Σ3. In the final section we pose a number of open problems. A set X with a topology ξ will be denoted by (X,ξ) and if p ∈ X, then ξ(p,X) denotes the collection of all open sets in X which contain p. The closure (respectively, interior) of a set A in a topological space (X,τ) will be denoted by clτ (A) (respectively, intτ (A)) or simply by cl(A) (respectively int(A)) when no confusion is possible. All undefined terms can be found in [7] or [13] and all spaces in this article are (at least) T3. A comprehensive survey of results on R-closed spaces and many open questions can be found in [8]. We make the following formal definitions. Definition 1.1. Say that two (distinct) T3-topologies τ1 and τ2 on a set X are adjacent in Σ3(X) if whenever σ ∈ Σ3(X) and τ1 ⊆ σ ⊆ τ2, then either σ = τ1 or σ = τ2. We say that τ1 is a lower topology in Σ3(X), τ2 is an upper topology in Σ3 and τ2 is an immediate successor of τ1. For a topology τ, τ + will always denote an immediate successor of τ. A T3-topology on X is R-minimal if there is no weaker T3-topology on X; it is well known that an R-minimal topology is R-closed. Clearly an R-minimal topology is not upper in Σ3(X). In the sequel, whenever the space X is understood, we will write Σ3 instead of Σ3(X). In [11], it was shown that the structure of basic intervals in Σ3 is essentially different from those of the poset Σt of Tychonoff spaces in that not every finite interval is isomorphic to the power set of a finite ordinal. The following result is Lemma 22 of [11]. Lemma 1.2. If σ is an immediate successor of τ in Σ3, then τ and σ differ at precisely one point. An open filter (that is, a filter with a base of open sets) F is a regular filter if for each U ∈ F there is V ∈ F such that cl(V ) ⊆ U. A simple application of Zorn’s Lemma shows that every regular filter can be embedded in a maximal regular filter and furthermore, in a regular space, if a maximal regular filter has an accumulation point, then it must converge to that point. By Theorem 4.14 of [4], a T3-space is R-closed if and only if every regular filter has an accumulation point or equivalently, if and only if every maximal regular filter converges. 2. Upper topologies in Σ3 The next result generalizes Theorem 2.14 of [1]. Regular topologies on a set 3 Theorem 2.1. Each T3-topology which is not R-closed is upper in Σ3. Proof. Suppose that (X,σ) is a T3-space which is not R-closed. Then there is some maximal regular filter F in (X,σ) which is not fixed. Pick p ∈ X and define a new topology τ on X as follows: τ = {U ∈ σ : p 6∈ U} ∪ {U ∈ σ : p ∈ U ∈ F}. The topologies τ and σ differ only at the point p and hence for each A ⊆ X, clτ (A) ⊆ clσ(A) ∪ {p}. We first show that (X,τ) is a T3-space; suppose that C ⊆ X is τ-closed and q 6∈ C. There are three cases to consider. 1) If p 6∈ C ∪{q}, then there are σ-open sets U,V separating C and q in X \{p} and U,V are τ-open. 2) If p ∈ C, then C is σ-closed and hence there are disjoint σ-open sets U and V such that C ⊆ U and q ∈ V . Furthermore, since F is a free regular filter, there is W ∈ F such that q 6∈ clσ(W) and hence q 6∈ clτ (W) = clσ(W) ∪ {p}. It is now clear that U ∪ W and U \ clτ (W) are disjoint τ-open sets containing C and q respectively. 3) If p = q, then since C is τ-closed and p 6∈ C, it follows that there is some element W ∈ F such that W ∩ C = ∅. Furthermore, since C is σ-closed, there are disjoint sets U,V ∈ σ such that C ⊆ U and p ∈ V . Since F is a regular filter, there is some T ∈ F such that clσ(T ) ⊆ W . Since clτ (T ) = clσ(T ) ∪ {p}, it is now clear that U \ clτ (T ) and V ∪ T are disjoint τ-open sets containing C and p respectively. We claim that τ is the immediate predecessor of σ in Σ3. To see this, suppose that µ is a T3-topology on X such that τ µ σ; note that µ differs from σ and τ only at the point p. If there is some µ-neighbourhood U of p which misses some element F ∈ F, then if W is a σ-open neighbourhood of p, it follows that W ∪ F is a τ-open, hence µ-open neighbourhood of p. But then (W ∪ F) ∩ U = W ∩ U ⊆ W is a µ-open neighbourhood of p, implying that µ = σ. Hence every µ-neighbourhood of p must meet every element of F; we claim that this implies that µ = τ. To prove our claim, let Vp be the filter of µ-open neighbourhoods of p and let G be the open filter generated by {F ∩ V : F ∈ F and V ∈ Vp}. We will show that G is a regular filter in (X,σ), thus contradicting the maximality of F. However, if F ∈ F and V ∈ Vp, then there is W ∈ Vp and H ∈ F such that V ⊇ clµ(W) ⊇ clσ(W) and clσ(H) ⊆ F . Hence W ∩ H ∈ G and clσ(W ∩ H) ⊆ F ∩ V . � In [6], the concept of a strongly upper topology was defined. (A topology τ is strongly upper if whenever µ τ, there is an immediate predecessor τ− of τ such that µ ⊆ τ− τ.) A simple modification of the above proof shows that every regular topology which is not R-closed is in fact strongly upper. Clearly every R-closed Tychonoff space is compact and hence the following result of [6] is an immediate corollary. 4 O. T. Alas and R. G. Wilson Corollary 2.2. Every Tychonoff topology which is not compact is (strongly) upper in Σ3. Every completely Hausdorff topology possesses a weaker Tychonoff topology (the weak topology induced by the continuous real-valued functions). Thus every completely Hausdorff R-minimal topology is compact. The following question then arises: Question 2.3. Is every completely Hausdorff T3-topology which is not compact an upper topology in Σ3? Question 2.4. Is every regular topology which has a compact Hausdorff subtop- ology an upper topology in Σ3? 3. Maximal R-closed topologies Recall that a space is submaximal if every dense set is open (we do not as- sume that a submaximal space must be dense-in-itself). It follows from 7M of [13] that each H-closed topology is contained in a maximal H-closed topology and that a space is maximal H-closed if and only if it is H-closed and submax- imal. However, as we show below, the class of submaximal R-closed spaces is much more restricted. Recall that a space is feebly compact if every locally finite family of open sets is finite. Theorem 3.1. Each submaximal regular, feebly compact topology has an iso- lated point. Proof. Suppose that (X,τ) is feebly compact, submaximal and has no isolated points. Fix p ∈ X and let C be a maximal cellular family of open sets in X so that for each C ∈ C, we have p 6∈ clτ (C). The subset ⋃ C is dense in X and hence F = X \ ⋃ C is closed and discrete. Since p ∈ F , there are disjoint open sets U and V so that p ∈ U and F \ {p} ⊆ V . Let S = {U ∩C : C ∈ C and U ∩ C 6= ∅}; since p is not isolated, it follows that S is an infinite cellular family of open sets. Since X is feebly compact, this family must have an accumulation point in F , and hence its only accumulation point is p. For each U ∩ C ∈ S, pick xC ∈ U ∩C; since X has no isolated points, the set {xC : U ∩C ∈ S}∪{p} is closed and discrete and hence there are disjoint opens sets U′ and V ′ such that p ∈ U′ and {xC : U ∩C ∈ S} ⊆ V ′. It follows immediately that the infinite family of non-empty open sets {C ∩ U ∩ V ′ : U ∩ C ∈ S} has no accumulation point in X, contradicting the fact that X is feebly compact. � The following theorem is a result of Scarborough and Stone [14]. For com- pleteness we include the simple proof. Regular topologies on a set 5 Theorem 3.2. An R-closed topology is feebly compact. Proof. Suppose to the contrary that U = {Un : n ∈ ω} is an infinite discrete family of open subsets of (X,τ). For each n ∈ ω, pick xn ∈ Un. It is then straightforward to check that the family B = {U ∈ τ : U ⊇ {xn : n ≥ k} for some k ∈ ω} is a regular filter base on X with no accumulation point, contradicting the fact that (X,τ) is R-closed. � Corollary 3.3. Each submaximal R-closed space has an isolated point. Lemma 3.4. An R-closed space which is scattered and of dispersion order 2 is compact. Proof. Suppose X = X0 ∪ X1 where X0 is the set of isolated points and X1 is the set of accumulation points of X. For each p ∈ X1, there is a closed neighbourhood U of p such that U ⊆ X0 ∪ {p}. It is clear that U is clopen and so X is 0-dimensional and hence Tychonoff. Thus X is compact. � Stephenson’s examples (see [15] and [9]) show that the previous result is false for R-closed scattered spaces of dispersion order 3. Since a subspace of a submaximal space is submaximal, the closure of the set of isolated points of an R-closed submaximal space is scattered of dispersion order 2. Thus: Corollary 3.5. Each submaximal R-closed space is a compact scattered space of dispersion order 2. Proof. Suppose that (X,τ) is an R-closed submaximal space and let X0 denote the set of isolated points of X; by Corollary 3.3, X0 6= ∅. Let C = cl(X \ cl(X0)); If C = ∅ then we are done, so suppose to the contrary. Then C is a submaximal space without isolated points and so again by Corollary 3.3, (C,τ|C) is not feebly compact. Thus there is an infinite locally finite family F of open sets in (C,τ|C). But then, {F ∩ (X \ cl(X0)) : F ∈ F} is an infinite locally finite family of open sets in X, implying that X is not feebly compact, which is a contradiction. � Theorem 3.6. A submaximal R-closed space is maximal R-closed. Proof. Suppose that (X,τ) is a submaximal R-closed space. By the previous corollary, X is compact scattered of dispersion order 2; let X0 denote the set of isolated points of X and X1 = X \ X0. Suppose that σ ! τ is a regular topology on X which differs from τ at a point p. Then there is some σ-open neighbourhood U of p which is not τ-open and hence does not contain any τ-neighbourhood of p; there is also a compact τ-neighbourhood V of p such that V ⊆ X0 ∪ {p}. It is then clear that V \ U is an infinite σ-closed subset of X0, implying that (X,σ) is not feebly compact. � 6 O. T. Alas and R. G. Wilson Lemma 3.7. A feebly compact regular space of countable pseudocharacter is first countable. Proof. Suppose that (X,τ) is feebly compact regular space and ψ(X,p) = ω. There is a family B = {Bn : n ∈ ω} of open sets such that ⋂ {Bn : n ∈ ω} = {p} and for each n ∈ ω, cl(Bn+1) ⊆ Bn. If B is not a local base at p, then there is some open neighbourhood U of p such that for each n ∈ ω, Bn * U. It is straightforward to check that the family of open sets {Bn \ (clτ (Bn+1 ∪ U)) : n ∈ ω} is an infinite locally finite family of open sets, contradicting the fact that X is feebly compact. � The next theorem should be compared with Theorem 2.20 of [12]. Theorem 3.8. A regular feebly compact first countable topology is maximal among regular feebly compact topologies. Proof. Suppose that (X,τ) is a regular feebly compact first countable space and σ ! τ is a regular topology on X; we will show that (X,σ) is not feebly compact. To this end, suppose that U ∈ σ \ τ; then X \ U is σ-closed but not τ-closed and so since (X,τ) is first countable, there is some sequence {pn} in X \ U convergent (in (X,τ)) to p ∈ U. By Lemma 4.1 of [2], there is a family of disjoint τ-open sets {Un : n ∈ ω} whose only accumulation point (in (X,τ)) is p and such that pn ∈ Un for each n ∈ ω. Now by regularity of (X,σ) there is W ∈ σ such that p ∈ W ⊆ clσ(W) ⊆ U; then, the collection of sets U = {Un \ clσ(W) : n ∈ ω} is a locally finite collection of open subsets of (X,σ) and so if an infinite number of elements of U are non-empty, then (X,σ) is not feebly compact. However, if for some n0 ∈ ω, Un \ clσ(W) = ∅ for all n ≥ n0, then pn ∈ Un ⊆ clσ(W) for all n ≥ n0 contradicting the fact that pn ∈ X \ U ⊆ X \ clσ(W). � The following result is now an immediate consequence of Theorems 3.2 and 3.8 and Lemma 3.7. Corollary 3.9. An R-closed space of countable pseudocharacter is maximal R-closed. Remark 3.10. Note that we have proved something a little stronger: If (X,τ) is R-closed and σ ⊇ τ differs from τ at a point of countable pseudocharacter, then (X,σ) is not R-closed. Corollary 3.11. A regular space with a strictly weaker R-closed first countable topology is upper in Σ3. Corollary 3.12. A first countable compact Hausdorff space is maximal R- closed. Question 3.13. Is a Fréchet compact Hausdorff space maximal R-closed? Regular topologies on a set 7 4. Lower topologies A point p is a maximal regular point of a regular space (X,τ) if the trace of the regular filter Vτp generated by τ(p,X) on X \ {p} is a maximal regular filter. Lemma 4.1. A point p in a regular topological space (X,τ) is a maximal regular point of X if and only if whenever τ σ is a regular topology on X such that σ|(X \ {p}) = τ|(X \ {p}) then p is an isolated point of (X,σ). Proof. For the sufficiency suppose that the regular filter Vτp generated by τ(p,X) when restricted to X \ {p} is not maximal. Then there is some regular filter F ! Vτp |(X \ {p}). Define σ to be that topology on X generated by the subbase τ ∪ {V ∪ {p} : V ∈ F}; it is straightforward to show that σ is a regular topology on X strictly finer that τ in which p is not an isolated point. To show the necessity, suppose that p is a maximal regular point of (X,τ). Then if σ ! τ and σ|(X \ {p}) = τ|(X \ {p}), it follows that the trace of the neighbourhood filter Vσp at p on X \ {p} is strictly larger than the trace of the neighbourhood filter Vτp at p on X \ {p} and since σ|(X \ {p}) = τ|(X \ {p}), Vσp |(X \ {p}) is a τ-open collection strictly larger than the maximal regular filter Vτp |(X \ {p}). It follows that p is an isolated point of (X,σ). � It was essentially shown in Theorem 2.13 of [1] that a point of first count- ability in a space is not a maximal regular point. Corollary 4.2. If (X,τ) has a maximal regular point then τ is a lower topology in Σ3. In [3] we characterized lower topologies in the poset of Hausdorff spaces as those having a closed subspace with a maximal point. Example 4.10 below shows that having a closed subspace with a maximal regular point does not guarantee that a topology is lower in Σ3. However, we have the following result: Lemma 4.3. If σ ∈ Σ3(X) is a simple extension of τ ∈ Σ3(X) which differs from τ at precisely one point p ∈ X, then σ is upper and each lower topology µ corresponding to σ has a closed subspace with a maximal regular point. Proof. It was shown in [6] that if a T3-topology σ is a simple extension of a T3-topology τ that differs from τ at precisely one point p, then σ is upper in Σ3(X) and is generated by the subbase τ ∪ {U ∪ {p}} for some U ∈ τ. Clearly µ∪{U ∪{p}} is also a subbase for σ and hence p is an isolated point of A = (X \U) ∪ {p} in the topology σ but not in µ. Thus p is a maximal regular point of (A,µ|A). � Remark 4.4. If τ is a lower topology in Σ3 and τ and τ + differ at p ∈ X then there is some U0 ∈ τ such that U0 ∪ {p} ∈ τ + \ τ. Then since τ+ is 8 O. T. Alas and R. G. Wilson regular, for each n ≥ 1 there is Un ∈ τ such that Un ∪ {p} ∈ τ + \ τ and Un ∪ {p} ⊆ clτ + (Un) ∪ {p} ⊆ Un−1 ∪ {p}. It is clear that τ + is generated by the subbase τ ∪ {Un ∪ {p} : n ∈ ω} and hence the character of p in (X,τ +) is no greater than its character in (X,τ). A family S = {Sn : n ∈ ω} is said to be strongly decreasing at p if for each n ∈ ω, cl(Sn+1) ∪ {p} ⊆ Sn ∪ {p}. We now formulate the above Remark as a lemma: Lemma 4.5. Let (X,τ) be a T3-space; if τ has an immediate successor τ + ∈ Σ3, then there is p ∈ X and a family U = {Un : n ∈ ω} ⊆ τ which is strongly decreasing at p, such that for each n ∈ ω, Un ∪ {p} 6∈ τ and τ + is generated by the subbase τ ∪ {Un ∪ {p} : n ∈ ω}. This result allows us to characterize (rather abstractly it must be said) lower topologies in Σ3 in the next theorem. In order to simplify the notation somewhat, when W = {Wn : n ∈ ω} ⊆ τ and V = {Vn : n ∈ ω} ∈ τ are strongly decreasing families at (a fixed) p ∈ X, τW will denote the topology generated by τ ∪ {Wn ∪ {p} : n ∈ ω} and W ∩ V will denote the family {Wn ∩Vn : n ∈ ω} which is also strongly decreasing at p. Theorem 4.6. A topology τ on X is lower in Σ3 if and only if there is p ∈ X and a strongly decreasing family U = {Un : n ∈ ω} ⊆ τ at p such that whenever V = {Vn : n ∈ ω} ⊆ τ is strongly decreasing at p and τU = τU∩V, then either τV = τU or τV = τ. Proof. Suppose that τ is not lower and fix p ∈ X; if U = {Un : n ∈ ω} ⊆ τ is strongly decreasing at p, then there is σ ∈ Σ3 such that τ σ τU . We may then choose a strongly decreasing family (at p) V = {Vn : n ∈ ω} ⊆ σ, such that for each n ∈ ω, Vn ∪ {p} ∈ σ \ τ and so τ τV τU . However, since for each n ∈ ω, Vn ∪ {p} ∈ τU , we have that (Un ∩ Vn) ∪ {p} ∈ τU which implies that τU = τU∩V , giving a contradiction. Conversely, suppose that τ is lower in Σ3; by Lemma 4.5, there is p ∈ X and a strongly decreasing family U at p such that τ+ = τU . Then, if V = {Vn : n ∈ ω} ⊆ τ is a strongly decreasing family at p such that τU = τU∩V it follows that for each n ∈ ω, Vn ∪ {p} ∈ τU and so τV ⊆ τU∩V = τU . � Theorem 4.7. A compact LOTS is maximal R-closed. Proof. Suppose that (X,τ,<) is a compact LOTS and σ ! τ. Then there is some U ∈ σ \ τ and p ∈ U such that U is not a τ-neighbourhood of p, and hence Lp \ U is cofinal in Lp \ {p} or Rp \ U is cofinal in Rp \ {p}, where Lp = {x ∈ X : x ≤ p} and Rp = {x ∈ X : x ≥ p}. It is easy to see that (X,τ) is maximal R-closed if and only if both of the compact subspaces (Lp,τ) and (Rp,τ) are maximal R-closed. Thus, if p is a point of first countability of (X,τ), then it is also of first countability in both (Lp,τ) and (Rp,τ) and so the result is an immediate consequence of Remark 3.10. Suppose then that χ(p,X) > ω, say χ(p,Lp) = κ > ω (where κ is a regular uncountable cardinal); in the sequel we consider only the subspace Lp. Let Regular topologies on a set 9 V ∈ σ be such that p ∈ V ⊆ clσ(V ) ⊆ U, then clearly, either, V = {p} or V \ {p} is a cofinal σ-closed subset of Lp \ {p}. If the former occurs, then clearly Lp \ {p} is open and closed in (Lp,σ) which then cannot be R-closed. If V \ {p} is cofinal in Lp \ {p} then, inductively we may construct interpo- lating sequences {vn : n ∈ ω} ⊆ V \ {p} and {wn : n ∈ ω} ⊆ Lp \ U such that wn < vn < wn+1 for all n ∈ ω. Since (X,<) is complete, q = sup{vn : n ∈ ω} = sup{wn : n ∈ ω} exists. Now for each n ∈ ω, let On = V ∩ (wn,wn+1). The sets {On : n ∈ ω} are σ-open and their only possible accumulation point in (X,σ) is q. There are now two possibilities: 1) If q ∈ clσ({wn : n ∈ ω}), then q ∈ Lp \ U and so q is not an accumulation point in (X,σ) of the family {On : n ∈ ω}, showing that (X,σ) is not feebly compact and hence not R-closed. 2) If on the other hand, q 6∈ clσ({wn : n ∈ ω}), then {wn : n ∈ ω}) is closed and discrete in (X,σ). Since σ is regular, we may construct a discrete family of σ-open sets {Wn : n ∈ ω} such that wn ∈ Wn, again showing that (X,σ) is not feebly compact. � The same proof essentially shows that: Theorem 4.8. If (X,τ,<) is a LOTS and χ(p,Lp) > ω, then p is a maximal regular point of Lp. Corollary 4.9. A compact LOTS is lower in Σ3 if and only if it is not first countable. Proof. The sufficiency follows from Theorem 4.8 and Corollary 4.2. The neces- sity was proved in Theorem 2.13 of [1]. � Compactness is essential in the previous theorem. It is straightforward to show that the one-point Lindelofication of a discrete space of cardinality ω1 is a LOTS but is neither first countable nor lower in Σ3. From Theorem 4.8 we see that if κ is an uncountable regular cardinal, then κ is a maximal regular point of κ + 1 (with the order topology). Example 4.10. Let κ denote the first ordinal of cardinality c+ and let X denote the set (κ + 1) × [0, 1], τ the product topology on X and σ the topology generated by τ ∪ {(κ, 1)}. We will show that σ = τ+. To this end, suppose that µ is a regular topology such that τ µ ⊆ σ; clearly µ differs from τ and σ only at the point (κ, 1) and hence there is some open µ-neighbourhood V which is not a τ-neighbourhood of (κ, 1) and some µ-neighbourhood U of (κ, 1) such that clµ(U) ⊆ V . Since κ > c, there are a number of possibilities: 1) There is an infinite set J = {rn : n ∈ ω} ⊆ [0, 1) with 1 ∈ cl(J) and for each n ∈ ω a set Sn ⊆ κ such that either, a) Sn is cofinal in κ or b) κ ∈ Sn and ⋃ {Sn × {rn} : n ∈ J} ∩ V = ∅. Or, 2) There is a cofinal set Sω ⊂ κ such that (Sω × {1}) ∩ V = ∅; furthermore, since V \ {(κ, 1)} is τ-open, we may assume that Sω is τ-closed in κ. 10 O. T. Alas and R. G. Wilson If 1a) occurs, then {κ} × J ⊆ X \ V ⊆ X \ clµ(U); and if 1b) occurs, then since V \ {(κ, 1)} is τ-open, it follows that {κ} × J ⊆ X \ V ⊆ X \ clµ(U). Thus in either case 1a) or 1b), there is an infinite subset J ⊆ [0, 1) with 1 ∈ cl(J) such that {κ} × J ⊆ X \ V ⊆ X \ clµ(U). It then follows that for each rn ∈ J there is αn ∈ κ such that ⋃ {(αn,κ] × {rn} : n ∈ J} ⊆ X \ clµ(U). Letting α = sup{αn : n ∈ J} ∈ κ we have that (α,κ] × J ⊆ X \ clµ(U) and so (α,κ) × {1} ⊆ X \ U. Again using regularity of (X,µ), there is some µ-open neighbourhood W of (κ, 1) such that clµ(W) ⊆ U and hence clµ(W) ∩ ((α,κ) × {1}) = ∅. If on the other hand, 2) occurs, then since clµ(U) is also τ-closed and it follows that clµ(U) ∩ (κ×{1}) is a τ-closed subset of κ×{1}). Thus, since κ is a regular cardinal with uncountable cofinality and clµ(U) ∩ Sω = ∅, it follows that there is some α ∈ κ such that clµ(U) ∩ ((α,κ) × {1}) = ∅. Thus in both cases 1) and 2) we have shown that there is a µ-open neigh- bourhood O of (κ, 1) and α ∈ κ such that clµ(O) ∩ ((α,κ) × {1}) = ∅. Now, since 1 ∈ cl(J), it follows that {(rn, 1] : n ∈ J} is a local base at 1 and so for each α < γ ∈ κ, there is rnγ ∈ J and Oγ open in κ such that Oγ × (rnγ , 1] ⊆ X \ clµ(O). Now denoting by Ln the set {γ : nγ = n ∈ J} and by Mn the set ⋃ {Oγ : γ ∈ Ln} we have that for each n ∈ J, Mn × (rn, 1] ⊆ X\clµ(O). However, ⋃ {Mn : n ∈ J} ⊇ (α,κ) and hence there is a finite subset {Mn1, . . . ,Mnk } which covers (α,κ). Letting r = max{rn1, . . . ,rnk }, we have that (α,κ) × (r, 1] ⊆ S \ clµ(O) and hence O ∩ ((α,κ + 1] × (r, 1]) ⊆ {κ} × [0, 1]. Since O∩(X\{(κ, 1)}) is τ-open this shows that O∩((α,κ+1]×(r, 1]) = {(κ, 1)}, that is to say, (κ, 1) is an isolated point of (X,µ). Of course, for each r ∈ [0, 1], the same argument applies to the point (κ,r) ∈ X. Thus each point of X is either a maximal regular point or a point of first countability; it follows that (κ + 1) × [0, 1] is maximal R-closed and is lower in Σ3. Now let L denote the ordered set (κ+ 1) ⊕ ω−1 (that is to say, κ+ 1 with its usual ordering followed by ω with its reverse ordering, with the order topology) and Y = L×[0, 1] with the product topology τ. The space Y is the product of two LOTS, is not first countable and contains X as a closed subspace. Nonetheless, we claim that Y is not lower in Σ3. To see this suppose that τ σ and that τ and σ differ at precisely one point p ∈ Y . By Theorem 2.13 of [1], p is not a point of first countability, hence p = (κ,r) ∈ {κ} × [0, 1]. Clearly the neighbourhood filter Vσp of p in (Y,σ) must differ from that in (Y,τ), V τ p , either on the subset (κ + 1) × [0, 1] or on Y \ (κ × [0, 1]). Suppose then that the traces of Vσp and V τ p on (κ + 1) × [0, 1] are the same; then V σ p and V τ p differ on Z = Y \ (κ × [0, 1]), however, (Z,τ) is first countable and hence again by Theorem 2.13 of [1] there are T3-topologies on it lying strictly between τ and σ. Thus τ and σ differ on (κ + 1) × [0, 1] and so by what we showed above, p must be an isolated point of ((κ + 1) × [0, 1],σ) and hence also of ({κ} × [0, 1],σ). However, the topology on Y \ (κ × [0, 1]) obtained by declaring {κ} × ([0, 1] \ {r}) to be closed is not regular, and an argument similar to that Regular topologies on a set 11 employed in Theorem 2.13 of [1] shows that there is no topology, minimal in the class of regular topologies larger than it. With a little more work, using the fact that [0, 1] is second countable, it is possible to substitute ω1 instead of κ in the previous example. However the following questions remain open. Question 4.11. If a regular topology is lower does some closed subspace have a maximal regular point? Question 4.12. Is there an internal concrete characterization of lower topolo- gies in Σ3? 5. First countable regular topologies Denote by Σ′3(X) the partially ordered set of first countable T3-topologies on a set X. Theorem 5.1. There are no jumps in Σ′3(X); between any two first count- able T3-topologies on X there are at least c incomparable first countable T3- topologies. Proof. Suppose that ξ and τ are two first countable T3-topologies on X which differ precisely at the point x ∈ X, Let {Vn : n ∈ ω} and {Wn : n ∈ ω} be nested local bases at x in the topologies ξ and τ respectively. We may now choose a sequence {xm}m∈ω which converges to x in (X,ξ) but not in (X,τ) and by passing to a subsequence if necessary, we may assume that xm ∈ Vm and {xm : m ∈ ω} is a closed, discrete subset of (X,τ). For each m ∈ ω, let {Unm : n ∈ ω} be a local base of τ-open sets at xm such that x 6∈ clτ (U n+1 m ) ⊆ Unm ⊆ Vm for each m,n ∈ ω; since (X,τ) is regular, we may assume that {U1m : m ∈ ω} is a discrete family of τ-open sets. Note that each set U n m is ξ-open and for each n ∈ ω, the family {Unm : m ∈ ω} has x as its unique accumulation point in (X,ξ). Now let A be an almost disjoint family of subsets of ω of size c and for each A ∈ A we define FA = {U ∈ τ : if x ∈ U then there is n ∈ ω and some finite F ⊆ ω such that U ⊇ ⋃ {Unm : m ∈ A \ F}}. It is clear that this is a sub-base for a first countable topology µA ⊆ τ on X and since {xm}m∈A converges to x in (X,µA) it follows that µA 6= τ. Furthermore, since Unm ⊆ Vm for each m,n ∈ ω, it follows that ξ ⊆ µA and since {xm}m∈ω\A does not converge to x in (X,µA) it follows that µA 6= ξ. Finally, note that if A,B ∈ A are distinct, then µA and µB are incomparable topologies. Finally, we need to show that each topology µA is regular. To this end, suppose that x ∈ U ∈ µA; then there is some finite set F ⊆ ω such that U ⊇ ⋃ {Unm : m ∈ A\F}}. It follows that ⋃ {clτ (U n+1 m ) : m ∈ A\F} is a µA-closed neighbourhood of x which is contained in U. If x 6= z ∈ U ∈ τ, then there is some τ-closed neighbourhood W ⊆ U of z and some n ∈ ω such that W ∩ ⋃ {Unm : m ∈ ω} = ∅ and hence W is a µA-closed neighbourhood of z contained in U. Thus (X,µA) is regular. � 12 O. T. Alas and R. G. Wilson In Theorem 2.13 of [1] it was shown that a sequential T3-topology of count- able pseudocharacter is not a lower topology in Σ3. However, we do not know the answer to the following question: Question 5.2. Is every first countable T3-topology which is not R-minimal, upper in Σ3? 6. Some more open problems The supremum of a chain of regular topologies is regular. Thus a positive answer to the first question would imply a positive answer to the second. Question 6.1. Is the supremum of a chain of R-closed topologies R-closed? Question 6.2. Is every R-closed topology contained in a maximal R-closed topology ? Note: There are maximal R-closed topologies which are not compact. In [15], Stephenson gave an example under CH of a first countable non-compact R- closed topology - by Corollary 3.9, this must be maximal R-closed. In [9] it was shown that the same construction can be done in ZFC. This space is scattered and has dispersion order 3. The topology contains a weaker compact Hausdorff topology of dispersion order 3 (which is clearly not maximal R-closed). Question 6.3. Is a maximal R-closed topology which is not R-minimal, upper in Σ3? Stephenson’s examples show that maximal R-closed topologies need not be lower. Finally, the most general question of all: Question 6.4. Is every regular topology which is not R-minimal an upper topology in Σ3? Acknowledgements. Research supported by Programa Integral de Fortalec- imiento Institucional (PIFI), grant no. 34536-55 (México) and Fundação de Amparo a Pesquisa do Estado de São Paulo (Brasil). The second author wishes to thank the Departament de Matemàtiques de la Universitat Jaume I for support from Pla 2009 de Promoció de la Investigació, Fundació Bancaixa, Castelló, during the preparation of the final version of this paper. Regular topologies on a set 13 References [1] O. T. Alas, S. Hernández, M. Sanchis, M. G. Tkachenko and R. G. Wilson, Adjacency in the partial orders of Tychonoff, regular and locally compact topologies, Acta Math. Hungar. 112, no. 3 (2006), 2005–2025. [2] O. T. Alas, M. G. Tkachenko and R. G. Wilson, Which topologies have immediate pre- decessors in the poset of Hausdorff topologies?, Houston Journal Math., to appear. [3] O. T. Alas and R. G. Wilson, Which topologies can have immediate successors in the lattice of T1-topologies?, Appl. Gen. Topol. 5, no. 2 (2004), 231–242. [4] M. Berri, J. Porter and R. M. Stephenson, A survey of minimal topological spaces, Proc. Kanpur Conference, 1968. [5] N. Carlson, Lower and upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007), 619–624. [6] C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008), 187–225. [7] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. [8] L. M. Friedler, M. Girou, D. H. Pettey and J. R. Porter, A survey of R-, U-, and CH- closed spaces, Topology Proc. 17 (1992), 71–96. [9] S. H. Hechler, Two R-closed spaces revisited, Proc. Amer. Math. Soc. 56 (1976), 303–309. [10] R. E. Larson and W. J. Thron, Covering relations in the lattice of T1-topologies, Trans. Amer. Math. Soc. 168 (1972), 101–111. [11] D. W. McIntyre and S. W. Watson, Finite intervals in the partial orders of zero- dimensional, Tychonoff and regular topologies, Topology Appl. 139 (2004), 23–36. [12] J. Porter, R. M. Stephenson and R. G. Woods, Maximal feebly compact spaces, Topology Appl. 52 (1993), 203–219. [13] J. Porter and R. G. Woods, Extensions and Absolutes of Topological Spaces, Springer Verlag, New York, 1987. [14] C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. [15] R. M. Stephenson, Two R-closed spaces, Canadian J. Math. 24 (1972), 286–292. [16] R. Valent and R. E. Larson, Basic intervals in the lattice of topologies, Duke Math. J. 379 (1972), 401–411. (Received October 2008 – Accepted September 2009) O. T. Alas (alas@ime.usp.br) Instituto de Matemática e Estat́ıstica, Universidade de São Paulo, Caixa Postal 66281, 05311-970 São Paulo, Brasil. R. G. Wilson (rgw@xanum.uam.mx) Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Postal 55-532, 09340, México, D.F., México. The structure of the poset of regular topologies on a set. By O. T. Alas and R. G. Wilson