On cellular-compact spaces
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ON CELLULAR-COMPACT SPACES 2 ´ 1,∗ , L. SOUKUP1 and Z. SZENTMIKLOSSY ´ I. JUHASZ 1
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Alfr´ ed R´ enyi Institute of Mathematics, Re´ altanoda u. 13-15, Budapest, Hungary e-mails: [email protected], [email protected]
Department of Analysis, E¨ otv¨ os University of Budapest, P´ azm´ any P´ eter s´ et´ any 1, Budapest, Hungary e-mail: [email protected] (Received December 16, 2019; revised January 21, 2020; accepted January 22, 2020)
Abstract. As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family U of non-empty open subsets of X there is a compact subspace K ⊂ X such that K ∩ U �= ∅ for each U ∈ U. In this note we answer several questions raised in [7] by showing that (1) any first countable cellular-compact T2 -space is T3 , and so its cardinality is at most c = 2ω ; (2) cov(M) > ω1 implies that every first countable and separable cellularcompact T2 -space is compact; (3) if there is no S-space then any cellular-compact T3 -space of countable spread is compact; (4) M Aω1 implies that every point of a compact T2 -space of countable spread has a disjoint local π-base.
1. Introduction A topological space X is said to be κ-cellular-compact if for any cellular family U of open subsets of X with |U| = κ there is a compact K ⊂ X such that K ∩ U �= ∅ for each U ∈ U . X is cellular-compact iff it is κ-cellularcompact for all cardinals κ. In [7, Theorem 4.13] the authors proved that the cardinality of a first countable cellular-compact T3 -space does not exceed the cardinality of the continuum and asked the natural question if this result can be extended to T2 -spaces: ∗ Corresponding
author. The research on and preparation of this paper was supported by OTKA grants no. K113047 and K129211. Key words and phrases: compact space, cellular-compact space, closed pseudocharacter, first countable space, weakly Lindel¨ of space. Mathematics Subject Classification: 54A25, 54A35, 54D30, 54D65. c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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´ ´ I. L. I. JUHÁSZ, JUHASZ, L. SOUKUP SOUKUP and and Z. Z. SZENTMIKLÓSSY SZENTMIKLOSSY
Question 5.1 [7]. Let X be a cellular-compact first countable T2 -space. Is it true that |X| ≤ 2ω ? In [1] a partial answer was given to this question by showing that this statement can be extended to the class of Urysohn spaces. In Theorem 2.7 we give the full affirmative answer by proving that, somewhat surprisingly, any first countable and cellular-compact T2 -space is actually T3 . In [7], under CH, a non-compact Tychonov cellular-compact space was constructed which is both first countable and separable. Consequently, the authors raised the following question: Question 5.2 [7]. Does there exist a model of ZFC in which every Tychonov cellular-compact space that is both separable and first countable is compact? Our Theorem 3.1 gives an affirmative answer to this question by showing that the assumption cov(M) > ω1 , or equivalently M Aω1 (co
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