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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\psi}_c(X)$$\end{document}, we show that |X|≤πχ(X)c(X)ψ¯c(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq \pi\chi(X)^{c(X)\overline{\psi}_c(X)}$$\end{document} for any topological space X. If X is Hausdorff then ψ¯c(X)≤ψc(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\psi}_c(X)\leq\psi_c(X)$$\end{document}; this gives a strengthening of a theorem of Shu-Hao [24]. We also prove that |X|≤2pwLc(X)t(X)pct(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq 2^{pwL_c(X)t(X)pct(X)}$$\end{document} for a homogeneous Hausdorff space X. The invariant pwLc(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$pwL_c(X)$$\end{document}, 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq 2^{L(X)t(X)pct(X)}$$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$pwL(X)\leq aL(X)$$\end{document}, demonstrating that 2pwL(X)χ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{pwL(X)\chi(X)}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G^c_\kappa$$\end{document}-sets lead to cardinality bounds involving the linear Lindelöf degree lL(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$lL(X)$$\end{document}, a weakening of L(X). It was shown in [5] that |X|≤2lL(X)F(X)ψ(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq 2^{lL(X)F(X)\psi(X)}$$\end{document} for Tychonoff spaces. We show the consistency of a) |X|≤2lL(X)F(X)ψc(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq 2^{lL(X)F(X)\psi_c(X)}$$\end{document} if X is Hausdorff, and b) |X|≤2lL(X)F(X)pct(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq 2^{lL(X)F(X)pct(X)}$$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|\leq 2^{L(X)t(X)pct(X)}$$\end{document} for homogeneous Hausdorff spaces.