() @ Appl. Gen. Topol. 19, no. 1 (2018), 155-161doi:10.4995/agt.2018.7888 c© AGT, UPV, 2018 Few remarks on maximal pseudocompactness Angelo Bella Department of Mathematics and Computer Science, University of Catania, Cittá universitaria viale A. Doria 6, 95125, Catania, Italy (bella@dmi.unict.it) Communicated by D. Georgiou Abstract A pseudocompact space is maximal pseudocompact if every strictly finer topology is no longer pseudocompact. The main result here is a counterexample which answers a question raised by Alas, Sanchis and Wilson. 2010 MSC: 54D55; 54D99. Keywords: pseudocompact; maximal pseudocompact; hereditarily maximal pseudocompact; accessible set. 1. Introduction For undefined notions we refer to [5] and [3]. Given a space X, we denote by τ(X) its topology. A Tychonoff space is pseudocompact if every real valued continuous function defined on it is bounded. Equivalently, a Tychonoff space X is pseudocompact if and only if every locally finite family of open sets is finite [5]. In a serie of papers Ofelia Alas, Richard Wilson and their co- authors have investigated the notion of maximal pseudocompactness (see [1], [2] and [8]). This notion is justified because a pseudocompact space can have a strictly finer Tychonoff topology which is still pseudocompact: consider for instance the compact space ω1+1 with the order topology. Indeed, let X be the space obtained by isolating the point ω1, i.e. X = ω1 ⊕{ω1}. As X is the topological sum of two countably compact Tychonoff spaces, it is pseudocompact and it clearly has a topology strictly finer than ω1 +1. A Tychonoff space (X, τ) is maximal pseudocompact Received 19 July 2017 – Accepted 07 October 2017 http://dx.doi.org/10.4995/agt.2018.7888 if (X, τ) is pseudocompact but (X, σ) is not pseudocompact for any Tychonoff topology σ strictly finer than τ. 2. Results An easy but useful fact is in the following: Lemma 2.1. Let (X, τ) be a T1 space and p ∈ X be a point of countable character. If σ is a Tychonoff topology on X such that σ ⊇ τ and (X, σ) is pseudocompact, then σ coincides with τ at p (i.e. p has the same system of neighbourhoods in both topologies). Proof. Fix a decreasing local base of open sets {Un : n < ω} at p in τ. If σ differs from τ at p, then there exists a closed neigbourhood V of p in σ which is not a neighbourhood of p in τ. But then, {Un \ V : n < ω} would be a locally finite family of non-empty open sets in (X, σ). This family is infinite because (X, τ) is T1 and we reach a contradiction. � Therefore, a first countable pseudocompact space is always maximal pseu- docompact. We begin by formulating a better sufficient condition. Let us say that a collection S hits a set A if S ∩ A 6= ∅ for each S ∈ S. A set S ⊆ X is co-pseudocompact [resp. co-singleton] if X \ S is pseudo- compact [resp. |X \ S| = 1]. Proposition 2.2. Let (X, τ) be a pseudocompact space and assume that for every co-pseudocompact set A ⊆ X and every point p ∈ A there exists a se- quence of open sets in X which hits A and converges to p. Then X is maximal pseudocompact. Proof. Assume by contradiction that there is a topology σ strictly finer than τ such that (X, σ) is still pseudocompact. If σ 6= τ at a point p, we may fix a regular closed neighbourhood V of p in σ which is not a neighbourhood of pin τ. The set X \ V is co-pseudocompact and p ∈ X \ V τ . Therefore, there exists a sequence {Un : n < ω} ⊆ τ converging to p and satisfying Un ∩ (X \ V ) 6= ∅ for every n. But then, {Un \ V : n < ω} would be a locally finite infinite family of open sets in (X, σ), in contrast with the pseudocompactness of σ. � The next observation shows that maximal pseudocompact spaces are “very close to” first countable. Proposition 2.3. If X is maximal pseudocompact, then each p ∈ X is the limit of a convergent sequence of non-empty open sets. Proof. Proposition 3.1 in [2] states that a maximal pseudocompact space has countable π-character, but the proof of this result actually establishes the stronger statement that every point is the limit of a convergent sequence of non-empty open sets. Indeed, let p be a non-isolated point of X. The maximal pseudocompactness of X implies that X \ {p} is not pseudocompact and so there is an infinite family of disjoint open sets {Un : n < ω} ⊆ X \ {p} which c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 156 Few remarks on maximal pseudocompactness is discrete in X \ {p}. We claim that the sequence {Un : n < ω} converges to p. If not, there would be an infinite set S ⊆ ω and a closed neighbourood V of p such that Un \ V 6= ∅ for each n ∈ S. But then, the infinite family of open sets {Un \ V : n ∈ S} would be discrete in X, in contrast with the pseudocompactness of X. � The above proposition shows that maximal pseudocompactness imposes strong conditions to the topology. Another non-trivial consequence is described in the following: Corollary 2.4. If X is maximal pseudocompat, then |X| ≤ 2c(X). Proof. By Proposition 3.1 in [2] and Šapirovskĭı’s formula w(X) ≤ πχ(X)c(X), there exists a dense set D such that |D| ≤ 2c(X). But, by Proposition 2.3 each point of X is the limit of a sequence contained in D and so |X| ≤ |D|ω ≤ 2c(X). � Thus, there are plenty of compact spaces which are not maximal pseudo- compact. In addition, by Corollary 3.5 in [2] every compactification of a non- compact pseudocompact space is not maximal pseudocompact. Propositions 2.2 and 2.3 seem to suggest that in the class of pseudocompact spaces maximal pseudocompactness (briefly MP) is a convergent-like property. Indeed, if P+ := “for every co-pseudocompact set A ⊆ X and every point p ∈ A there exists a sequence of open sets in X which hits A and converges to p” and P− := “every point is the limit of a converging sequence of non-empty open sets”, then pseudocompact + P+ =⇒ MP =⇒ P− The one-point compactification of an uncountable discrete space is a compact space satisfying P+ which is not first countable. We believe there should exist a maximal pseudocompact space which does not satisfy P+, but at moment we do not have such a space. On the other direction, let X = A ∪ ω be a Ψ-space over a MAD family A on ω and let X ∪ {∞} be its one-point compactification. Fix A0 ∈ A and let Z be the quotient space of X ∪{∞} obtained by identifying A0 and ∞ to a point p. Z is a compact space which satisfies P −. This is evident for each point of Z \ {p}. For p observe that {{n} : n ∈ A0} is a sequence of open sets in Z converging to p. But, Z is not maximal pseudocompact, because the function f : X → Z, defined by letting f(A0) = p and f(x) = x for every x ∈ X \ {A0}, is a continuous bijection which is not open. We conclude that the unknown property P which characterizes maximal pseudocompactness within the class of pseudocompact space lies in between P+ and P− and differs from the latter. Since P− is just P+ restricted to co-singleton sets (a subclass of co-pseu- docompact sets), property P should involve an appropriate subclass of co- pseudocompact sets. Question 2.5. What is the convergent property P such that a pseudocompact space X is maximal pseudocompact if and only if X satisfies P? c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 157 As pointed out in [1], a relevant role in studying maximal pseudocompactness is played by the notion of accessibility from a dense subset. Given a space X and a dense set D ⊆ X, we say that X is strongly accessible from D if for any x ∈ X \ D and any A ⊆ D such that x ∈ A, there exists a countable sequence S ⊆ A converging to x. Proposition 2.6 ([1, Theorem 2.4]). Let X be a pseudocompact space and D a dense set of isolated points. If X is strongly accessible from D, then X is maximal pseudocompact. Therefore, any compactification γ(N) of the set of integers N with the dis- crete topology such that γ(N) is strongly accessible from N is maximal pseu- docompact [1]. In some case, γ(N) \ N can be homeomorphic to ω1 + 1, thus showing for instance that a compact maximal pseudocompact space need not be Fréchet. The first construction of this kind, discovered in the attempt to find a com- pact radial separable non Fréchet space, is the space δ(N) given in [7] by assum- ing the Continuum Hypothesis. A similar example, obtained under the weaker assumption d = ω1, is given in [6]. But perhaps, the easiest way to obtain it is by using a tower. Recall that the cardinal t is the smallest size of a tower, i.e. a well-ordered by reverse almost inclusion family of subsets of N without any infinite pseu- dointersection (see [3] for more). Fix a family A = {Aα : α ∈ ω1} of subsets of N, well-ordered by ⊂ ∗. Furthermore, put A−1 = ∅ and Aω1 = N. We define a topology on the set γ(N) = N∪ω1 +1 by declaring each point of N isolated and by taking as a local base at each α ∈ ω1 +1 the sets ]β, α]∪Aα \ (Aβ ∪F), where F is a finite subset of N and −1 ≤ β < α. To be more formally correct, we should replace in the previous definition ω1 + 1 for instance with {xα : α ∈ ω1 + 1}. However, we believe our semplified notation does not cause any trouble to the reader. The space γ(N) is compact Hausdorff and first countable at each α < ω1. Proposition 2.7. The space γ(N) may fail to be maximal pseudocompact if and only if t = ω1. Proof. We begin by showing that t > ω1 implies the maximal pseudocompact- ness of γ(N). By Proposition 2.6, it suffices to check that γ(N) is strongly accessible from N. As the only point of uncountable character in γ(N) is ω1, we only need to consider the case of a set A ⊆ N such that ω1 ∈ A. This clearly implies |(N \ Aα) ∩ A| = ω for each α. Since t > ω1, the family {(N \ Aα) ∩ A : α < ω1} is not a tower and hence we may take an infinite set S ⊆ A satisfying S ⊆∗ N \ Aα for every α < ω1. This actually means that S converges to ω1 and we are done. To complete the proof, we now verify that t = ω1 implies that γ(N) may fail to be maximal pseudocompact. The point is that, by assuming t = ω1, we may choose the family A in such a way that {N \ Aα : α < ω1} is a tower. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 158 Few remarks on maximal pseudocompactness Consequently, if we take an infinite set S ⊆ N, then there exists some α < ω1 such that |S ∩ Aα| = ω. If α is the least ordinal with this property, then the set S ∩ Aα is actually a sequence converging to α. Therefore, no subsequence of N can converge to ω1. In particular, the point ω1 cannot be the limit of a sequence of non-empty open subsets of γ(N) and so by Proposition 2.3 the space γ(N) is not maximal pseudocompact. � The above discussion suggests the following: Question 2.8 (ZFC). Is there a compactification γ(N) of N which is maximal pseudocompact but not Fréchet? Any radial compactification of N is obviously maximal pseudocompact, but Dow [4] has shown that there are models where every compact separable radial space is Fréchet. In [1], Question 2.17 asks whether Proposition 2.6 is reversible, namely: “Suppose that X is a maximal pseudocompact space with a dense set of isolated points D. Is X strongly accessible from D?” Mimiking the construction given in [1], Example 2.14, we can give a consis- tent negative answer to the above question. Example 2.9. [t > ω1] or [d = ω1] A maximal pseudocompact space which is not accessible from a dense set of isolated points. Proof. Take the space γ(N) = N∪ω1 +1 described above under the assumption t > ω1 or d = ω1and let Y = (ω + 1) × ω1. Let X be the quotient space of Y ⊕ γ(N), obtained by identifying the set {ω} × ω1 ⊆ Y with the copy of ω1 in γ(N) (i.e. (ω, α) ≡ α for each α ∈ ω1). Recall that if q : Y ⊕ γ(N) → X is the quotient map, then V ∈ τ(X) if and only if q−1(V ) ∈ τ(Y ⊕ γ(N)) if and only if q−1(V ) ∩ Y ∈ τ(Y ) and q−1(V ) ∩ γ(N) ∈ τ(γ(N)). We claim that X is maximal pseudocompact. Indeed, let σ be a pseudocom- pact topology finer than the topology τ on X. Since γ(N) is compact, q ↾ γ(N) is an embedding and so we may identify γ(N) with q(γ(N)) ⊆ X. We must have ω1 ∈ N σ , otherwise a sequence S ⊆ N, converging in τ to ω1, would provide a discrete infinite family of open singletons in σ. Since by Lemma 2.1 σ coincides with τ at each α ∈ ω1, we see that N σ = γ(N). Thus γ(N) is regular closed in σ and hence pseudocompact in σ. Since γ(N) is maximal pseudocompact, we conclude that σ coincides with τ on γ(N), i.e. σ ↾ γ(N) = τ ↾ γ(N). Since X is first countable at each point of q(Y ), again by Lemma 2.1, σ coincides with τ on q(Y ), so we also have σ ↾ q(Y ) = τ ↾ q(Y ). Now, take any V ∈ σ. From V ∩ q(Y ) ∈ σ ↾ q(Y ) = τ ↾ q(Y ), it follows q−1(V ) ∩ Y ∈ τ(Y ). In a similar manner, from V ∩ γ(N) ∈ σ ↾ γ(N) = τ ↾ γ(N) = τ ↾ q(γ(N)), it follows q−1(V ) ∩ γ(N) ∈ τ(γ(N)). This suffices to conclude that V ∈ τ and hence σ = τ(X). X has a dense set of isolated points, namely D = (ω × {0, α + 1 : α < ω1})∪N. But, X is not strongly accessible from D, because ω1 is in the closure of ω × {α + 1 : α < ω1}, but no subsequence of it can converge to ω1. This last thing depends on the countable compactness of Y . � c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 159 In [1] a space X was called hereditarily maximal pseudocompact (briefly HMP) if each closed subspace of X is maximal pseudocompact. Since pseudocompactness is preserved by passing to regular closed subspaces, the following way to define the hereditary version of maximal pseudocompact- ness certainly makes sense. A space is weakly hereditarely maximal pseudocompact (briefly wHMP) if every regular closed subspace is maximal pseudocompact. Example 2.14 in [1] as well as the space X in Example 2.9 above provide maximal pseudocompact spaces which are not wHMP. Regarding Example 2.9, observe that the set q(ω × ω1) is open in X and q(ω × ω1) is homeomorphic to Y ∪{ω1}. But the latter is a regular closed subspace of X which is not maximal pseudocompact. A non-trivial difference between HMP and wHMP emerges from the follow- ing: Proposition 2.10. If the pseudocompact space X is strongly accessible from a dense set D of isolated points, then X is wHMP. Proof. Let Y be a regular closed subset of X. Since Y = U for some open set U, Y is strongly accessible from the dense subset of isolated points U ∩ D and we are done. � Any version of the space γ(N), mentioned above, is therefore wHMP but not HMP. In connection with Question 2.17 in [1], consider the following: Proposition 2.11. Let X be a wHMP space. If D is a dense set of isolated points, then X is strongly accessible from D. Proof. Take a point x ∈ X \ D and a set A ⊆ D such that x ∈ A. Since A is open, we see that the subspace A is maximal pseudocompact. Therefore, by Proposition 2.3 (or Lemma 2.8 in [1]), there is a sequence in A converging to x. � Propositions 2.10 and 2.11 show that Proposition 2.6 is reversible precisely for wHMP spaces in the class of pseudocompact spaces. Acknowledgements The author thanks Richard Wilson for the useful discussion concerning the presentation of Example 2.9. A great thank also to the referee for the useful comments and the careful reading. This research was partially supported by a grant of the group GNSAGA of INdAM. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 160 Few remarks on maximal pseudocompactness References [1] O. T. Alas, M. Sanchis and R. G. Wilson, Maximal pseudocompact and maximal R- closed spaces, Houston J. Math. 38 (2012), 1355–1367. [2] O. T. Alas, W. W. Tkachuk and R. G. Wilson, Maximal pseudocompact spaces and the Preiss-Simon property, Central Eur. J. Math. 12 (2014), 500–509. [3] E.K. van Douwen, The integers and topology, in: Handbook of Set-theoretic Topology (K. Kunen and J. E. Vaughan Editors), Elsevier Science Publishers B.V., Amsterdam, (1984), 111–160. [4] A. Dow, On compact separable radial spaces, Canad. Math. Bull. 40 (1997), 422–432. [5] R. Engelking, General topology, Heldermann Verlag, Berlin (1989). [6] P. J. Nyikos, Convergence in topology, in: Recent Progress in General Topology. Elsevier Science Publishers B. V. (Amsterdam). M. Husek and J. van Mill ed. (1992), 537–570. [7] K. M. Devi, P. R. Meyer and M. Rajagopalan, When does countable compactness imply sequential compactness?, General Topology Appl. 6 (1976), 279–289. [8] V. V. Tkachuk and R. G. Wilson, Maximal countably compact spaces and embeddings in MP spaces, Acta Math. Hungar. 145, no. 1 (2015), 191–204. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 161