Cardinality bounds via covers by compact sets

We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant ψ¯ c(X) , we show that |X|≤πχ(X)c(X)ψ¯c(X) for any topological space X. If X is Hausdorff then ψ¯ c(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| ≤ 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) ≤ aL(X) , demonstrating that 2 pwL(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 Gκc-sets lead to cardinality bounds involving the linear Lindelöf degree lL(X) , a weakening of L(X). It was shown in [5] that | X| ≤ 2 lL(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| ≤ 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| ≤ 2 L(X)t(X)pct(X) for homogeneous Hausdorff spaces.