Cardinality bounds via covers by compact sets
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CARDINALITY BOUNDS VIA COVERS BY COMPACT SETS A. BELLA1,† and N. CARLSON2,∗ 1
Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy e-mail: [email protected] 2
Department of Mathematics, California Lutheran University, 60 W. Olsen Rd, MC 3750, Thousand Oaks, CA 91360, USA e-mail: [email protected] (Received June 7, 2020; revised August 21, 2020; accepted August 22, 2020)
Abstract. We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant ψc (X), we show that |X| ≤ πχ(X)c(X)ψc (X) for any topological space X. If X is Hausdorff then ψc (X) ≤ ψc (X); this gives a strengthening of a theorem of Shu-Hao [24]. We also prove that |X| ≤ 2pwLc (X)t(X)pct(X) for a homogeneous Hausdorff space X. The invariant pwLc (X), introduced in [9], is bounded above by both L(X) and c(X). Our result thus improves the bound |X| ≤ 2L(X)t(X)pct(X) for homogeneous Hausdorff spaces X [13] and represents a new extension of de la Vega’s Theorem [15] into the Hausdorff setting. Moreover, we show pwL(X) ≤ aL(X), demonstrating that 2pwL(X)χ(X) is not a cardinality bound for all Hausdorff spaces. This answers a question of Bella and Spadaro [9]. A further theorem on covers by Gcκ -sets lead to cardinality bounds involving the linear Lindel¨ of degree lL(X), a weakening of L(X). It was shown in [5] that |X| ≤ 2lL(X)F (X)ψ(X) for Tychonoff spaces. We show the consistency of a) |X| ≤ 2lL(X)F (X)ψc (X) if X is Hausdorff, and b) |X| ≤ 2lL(X)F (X)pct(X) if X is Hausdorff and homogeneous. If X is additionally regular, the former consistently improves the result from [5]. The latter gives a consistent improvement of the inequality |X| ≤ 2L(X)t(X)pct(X) for homogeneous Hausdorff spaces.
1. Introduction In this study we formulate several theorems concerning covers of topological spaces by compact and related sets. These in turn generate bounds ∗ Corresponding † The
author. first author wishes to thank the University of Catania for the grant PIACERI 2020/22,
Linea 2. Key words and phrases: ardinality bound, cardinal invariant, countably tight space, homogeneous space, weak tightness. Mathematics Subject Classification: 54A25.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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A. A. BELLA and N. CARLSON
on the cardinality of a space X. This is achieved by either a) considering the singleton cover {{x} : x ∈ X} or b) demonstrating the existence of a compact set K with certain properties and covering X via homeomorphic images of K, as can be done if the space is homogeneous. Recall that a space X is homogeneous if for all x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. Shu-Hao [24] gave an improvement of the Hajnal-Juh´ asz inequality by c(X)ψ c (X) showing that |X| ≤ πχ(X) for a Hausdorff space X. (Previously ˇ Sapirovski˘ ı [23] had shown |X| ≤ πχ(X)c(X)ψ(X) for a regular space X.) In Corollary 2.3 we give an improvement of Shu-Hao’s result by demonstrating that ψc (X) can
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