Cardinality bounds involving the skew-λ Lindelöf degree and its variants

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)skL(X,λ), the skew-λ Lindelöf degree of a space X, where λ   is a cardinal. skL(X,λ)skL(X,λ) is a weakening of the Lindelöf degree and is defined as the least cardinal κ   such that if UU is an open cover of X   then there exists V∈[U]≤κV∈[U]≤κ such that |X\∪V|<λ|X\∪V|<λ. We show that if X   is Hausdorff then |X|≤2skL(X,λ)t(X)ψ(X)|X|≤2skL(X,λ)t(X)ψ(X), where λ=2t(X)ψ(X)λ=2t(X)ψ(X). This improves the well-known Arhangel'skiĭ–Šapirovskiĭ bound 2L(X)t(X)ψ(X)2L(X)t(X)ψ(X) for the cardinality of a Hausdorff space X  . We additionally define several variations of skL(X,λ)skL(X,λ), establish other related cardinality bounds, and provide examples.