Cardinality bounds via covers by compact sets

Abstract

We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant $$\overline{\psi}_c(X)$$ , we show that $$|X|\leq \pi\chi(X)^{c(X)\overline{\psi}_c(X)}$$ for any topological space X. If X is Hausdorff then $$\overline{\psi}_c(X)\leq\psi_c(X)$$ ; this gives a strengthening of a theorem of Shu-Hao [24]. We also prove that $$|X|\leq 2^{pwL_c(X)t(X)pct(X)}$$ for a homogeneous Hausdorff space X. The invariant $$pwL_c(X)$$ , introduced in [9], is bounded above by both L(X) and c(X). Our result thus improves the bound $$|X|\leq 2^{L(X)t(X)pct(X)}$$ 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 $$pwL(X)\leq aL(X)$$ , demonstrating that $$2^{pwL(X)\chi(X)}$$ is not a cardinality bound for all Hausdorff spaces. This answers a question of Bella and Spadaro [9]. A further theorem on covers by $$G^c_\kappa$$ -sets lead to cardinality bounds involving the linear Lindelof degree $$lL(X)$$ , a weakening of L(X). It was shown in [5] that $$|X|\leq 2^{lL(X)F(X)\psi(X)}$$ for Tychonoff spaces. We show the consistency of a) $$|X|\leq 2^{lL(X)F(X)\psi_c(X)}$$ if X is Hausdorff, and b) $$|X|\leq 2^{lL(X)F(X)pct(X)}$$ 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 $$|X|\leq 2^{L(X)t(X)pct(X)}$$ for homogeneous Hausdorff spaces.