Cardinality bounds involving the skew-$λ$ 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 $skL(X,λ)$, the skew-$λ$ Lindelöf degree of a space $X$, where $λ$ is a cardinal. $skL(X,λ)$ is a weakening of the Lindelöf degree and is defined as the least cardinal $κ$ such that if $\mathcal{U}$ is an open cover of $X$ then there exists $\mathcal{V}\in [\mathcal{U}]^{\leqκ}$ such that $|X\backslash\cup\mathcal{V}|<λ$. We show that if $X$ is Hausdorff then $|X|\leq 2^{skL(X,λ)t(X)ψ(X)}$, where $λ= 2^{t(X)ψ(X)}$. This improves the well-known Arhangel'skii- Šapirovskii bound $2^{L(X)t(X)ψ(X)}$ for the cardinality of a Hausdorff space $X$. We additionally define several variations of $skL(X,λ)$, establish other related cardinality bounds, and provide examples.