2015/07/23 by Nathan Carlson, Carlson, Nathan, Jack Porter +1
Mathematics · #54A25 #54D10 #54D20 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #math.GN #msc:54A25 #msc:54D10 #msc:54D20
paper · pdf · doi:10.48550/arxiv.1507.06684
arxiv created 2015/07/23 · openalex publication_date 2015/07/23 · arxiv updated 2015/07/27 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
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 U is an open cover of X then there exists V∈ [U]≤κ such that |X\backslash\cupV|<λ. We show that if X is Hausdorff then |X|≤ 2skL(X,λ)t(X)ψ(X), where λ= 2t(X)ψ(X). This improves the well-known Arhangel'skii- Šapirovskii bound 2L(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.