Cardinality bounds involving the skew-$\lambda$ Lindelöf degree and its variants

Abstract

We introduce a modified closing-off argument that results in several improved bounds for the cardinalities of Hausdorff and Urysohn spaces. These bounds involve the cardinal invariant $skL(X,\lambda )$, the skew-$\lambda$ Lindelöf degree of a space $X$, where $\lambda$ is a cardinal. $skL(X,\lambda )$ is a weakening of the Lindelöf degree and is defined as the least cardinal $\kappa$ such that if $\mathcal{U}$ is an open cover of $X$ then there exists $\mathcal{V}\in [\mathcal{U}{]}^{\le \kappa }$ such that $|X\backslash \cup \mathcal{V}|<\lambda$. We show that if $X$ is Hausdorff then $|X|\le 2^{skL(X,\lambda )t(X)\psi (X)}$, where $\lambda =2^{t(X)\psi (X)}$. This improves the well-known Arhangel'skii- Šapirovskii bound $2^{L(X)t(X)\psi (X)}$ for the cardinality of a Hausdorff space $X$. We additionally define several variations of $skL(X,\lambda )$, establish other related cardinality bounds, and provide examples.