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Selective Sequential Pseudocompactness

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Selective Sequential Pseudocompactness SELECTIVE SEQUENTIAL PSEUDOCOMPACTNESS ALEJANDRO DORANTES-ALDAMA AND DMITRI SHAKHMATOV Dedicated to Professor Filippo Cammaroto on the occasion of his 65th anniversary Abstract. We say that a topological space X is selectively sequentially pseudocompact if for every family {Un : n ∈ N} of non-empty open subsets of X, one can choose a point xn ∈ Un for every n ∈ N in such a way that the sequence {xn : n ∈ N} has a convergent subsequence. We show that the class of selectively sequentially pseudocompact spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc´ıa-Ferreira and Ortiz- Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every ω-bounded (=the closure of which countable set is compact) group is selectively sequentially pseudocompact, while compact spaces need not be selectively sequentially pseudocompact. Finally, we construct selectively sequentially pseudocompact group topologies on both the free group and the free Abelian group with continuum-many generators. All topological spaces are considered to be Tychonoff and all topological groups Hausdorff. The symbol N denotes the set of natural numbers, R denotes the set of real numbers, T denotes the circle group {eiθ : θ ∈ R}⊆ R2. If f : X → Y is a function and A is a subset of Y , then we let f ←(A)= {x ∈ X : f(x) ∈ A}. For a subset A of a topological space X, we use clX (A) to denote the closure of A in X. Let us recall two well-known compactness-type properties. Definition 0.1. A topological space X is called: (i) sequentially compact if every sequence in X has a convergent subsequence; (ii) pseudocompact if every continuous real-valued function defined on X is bounded; The notion of sequential compactness is well known; see, for example, [9, Section 3.10]. The notion of pseudocompactness is due to Hewitt [13]. Definition 0.2. We say that an indexed sequence {Un : n ∈ N} is: • point-finite provided that the set {n ∈ N : x ∈ Un} is finite for each point x ∈ X; • locally finite provided that each point x ∈ X has an open neighborhood W such that the set N(W )= {n ∈ N : W ∩ U 6= ∅} is finite. arXiv:1702.02055v1 [math.GN] 7 Feb 2017 n Clearly, every locally finite sequence is point-finite. The following fact is well known. Fact 0.3. For a topological space X, the following conditions are equivalent: (i) X is pseudocompact; (ii) every sequence of non-empty open subsets of X is not locally finite; (ii) every sequence of pairwise disjoint non-empty open subsets of X is not locally finite. 2010 Mathematics Subject Classification. Primary: 54D30; Secondary: 22A05, 22C05, 54A20, 54H11. Key words and phrases. convergent sequence, sequentially compact, pseudocompact, strongly pseudocompact, sequentially pseudocompact, topological group, free abelian group, variety of groups. The first listed author was supported by CONACyT of M´exico: Estancias Posdoctorales al Extranjero propuesta No. 263464. The second listed author was partially supported by the Grant-in-Aid for Scientific Research (C) No. 26400091 by the Japan Society for the Promotion of Science (JSPS). 1 2 A.DORANTES-ALDAMAANDD.SHAKHMATOV 1. Disjoint refinement principle The next lemma is probably known. We include its proof here only for the reader’s convenience. Lemma 1.1. Let U = {Un : n ∈ N} be a point-finite family of non-empty open subsets of a topological space X which is not locally finite. Then there exist a sequence V = {Vk : k ∈ N} of pairwise disjoint non-empty open subsets of X and a strictly increasing sequence {nk : k ∈ N}⊆ N N such that Vk ⊆ Unk for all k ∈ . Proof. Since U is not locally finite, there exists x ∈ X such that (1) the set N(W )= {n ∈ N : W ∩ Un 6= ∅} is infinite for every open neighborhood W of x. Since U is point-finite, we can fix n−1 ∈ N such that (2) x 6∈ Un for every n ∈ N satisfying n ≥ n−1. By induction on k ∈ N we shall select nk ∈ N and a non-empty open subset Vk of X satisfying the following conditions: (ik) nk−1 <nk, (iik) clX (Vk) ⊆ Unk , (iiik) if k ≥ 1, then Vk ∩ Vj = ∅ whenever j ∈ N and j < k. We let n0 = n−1 + 1 and use regularity of X to fix a non-empty open subset V0 of Un0 satisfying (ii0). Condition (i0) is clear and condition (iii0) is vacuous. Suppose now that k ∈ N, k ≥ 1 and nk−1 ∈ N and a non-empty open subset Vk−1 of X satisfying (ik−1)–(iiik−1) have already been chosen. Let j ∈ N and j < k. Since (il) holds for all l ∈ N with l ≤ j, we have n−1 < n0 < · · · < nj. Combining this with (2), we conclude that x 6∈ Unj . Finally, from this and (iij), we get x 6∈ clX (Vj). N Since this holds for every j ∈ with j < k, it follows that W = X \ j<k clX (Vj) is an open neighborhood of x. Since the set N(W ) is infinite by (1), we can chooseSnk ∈ N(W ) satisfying nk−1 <nk. Then W ∩ Unk 6= ∅ by the definition of N(W ), so we can choose a point z ∈ W ∩ Unk and its open neighborhood Vk such that clX (Vk) ⊆ Unk . Clearly, conditions (ik)–(iiik) are satisfied. The inductive step has been completed. The sequence {nk : k ∈ N} is strictly increasing, as (ik) holds for every k ∈ N. Since (iiik) holds for all k ∈ N, we conclude that V = {Vk : k ∈ N} is a sequence of non-empty pairwise disjoint open subsets of X. Finally, Vk ⊆ Unk by (iik). Definition 1.2. Let X be a topological space. (i) UX denotes the family of all sequences U = {U(n) : n ∈ N} of non-empty open subsets of X. (ii) We say that U ∈ UX is pairwise disjoint if U(n)∩U(m)= ∅ whenever m,n ∈ N and m 6= n. (iii) If U ∈ UX and s : N → N is a monotonically increasing function, then the function U ◦ s ∈ UX shall be called a subsequence of U. Proposition 1.3. Let X be a pseudocompact space and let P : UX → {0, 1} be a function satisfying the following conditions: (a) P(U) = 1 whenever U ∈ UX is not point-finite; (b) if P(V ) = 1 for some subsequence V of U ∈ UX , then P(U) = 1. (c) if U, V ∈ UX , V (n) ⊆ U(n) for every n ∈ N and P(V ) = 1, then P(U) = 1. (d) if V ∈ UX is pairwise disjoint, then P(V ) = 1. Then P(U) = 1 for all U ∈ UX . Proof. Let U ∈ UX be arbitrary. By (a), we may assume, without loss of generality, that U is point-finite. Since X is pseudocompact, U cannot be locally finite by Fact 0.3. By Lemma 1.1, SELECTIVESEQUENTIALPSEUDOCOMPACTNESS 3 there exist a pairwise disjoint V ∈ UX and a monotonically increasing function s : N → N such that (3) V (k) ⊆ U(s(k)) = (U ◦ s)(k) for every k ∈ N. Since V ∈ UX is pairwise disjoint, P(V ) = 1 by (d). From this, (3) and (c), we get P(U ◦ s) = 1. Since U ◦ s is a subsequence of U, (b) implies P(U) = 1. Corollary 1.4. Let X be a pseudocompact space and let P : UX → {0, 1} be a function satisfying conditions (a), (b), (c) from Proposition 1.3. Then the following conditions are equivalent: (i) P(U) = 1 for every U ∈ UX ; (ii) P(V ) = 1 for each pairwise disjoint V ∈ UX . We demonstrate the usefulness of this corollary on the following property which is perhaps not so familiar to the reader. Definition 1.5. A topological space X is called sequentially pseudocompact if for every family {Un : n ∈ N} of non-empty open subsets of X, there exists an infinite set J ⊆ N and a point x ∈ X such that the set {n ∈ J : W ∩ Un = ∅} is finite for every open neighborhood W of x. This notion appeared in [8, Definition 1.4] under the name sequentially feebly compact. A formally weaker property obtained by requiring the conclusion of Definition 1.5 to hold only for the families {Un : n ∈ N} consisting of pairwise disjoint non-empty open subsets of X was defined earlier in [3, Definition 1.8]. It was proved in [15, Proposition 1] that these two versions of Definition 1.5 are in fact equivalent. This result can be easily derived from the disjoint refinement principle. Corollary 1.6. [15, Proposition 1] For every topological space X, the following conditions are equivalent: (i) X is sequentially pseudocompact; (ii) for every sequence {Un : n ∈ N} of pairwise disjoint non-empty open subsets of X, there exists an infinite set J ⊆ N and a point x ∈ X such that the set {n ∈ J : W ∩ Un = ∅} is finite for every open neighborhood W of x. Proof. The implication (i)⇒(ii) is trivial. Let us prove that (ii)⇒(i). Note that (ii) implies that every pairwise disjoint sequence of non-empty open subsets of X is not locally finite. Therefore, X is pseudocompact by Fact 0.3. Define the function P : UX → {0, 1} as follows. For U ∈ UX , we let P(U) = 1 if there exists an infinite set J ⊆ N and a point x ∈ X such that the set {n ∈ J : W ∩ U(n)= ∅} is finite for every open neighborhood W of x, and we define P(U) = 0 otherwise.
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