2023/09/26 by Nathan Carlson, Carlson, Nathan
Decision Sciences · Mathematics · #54A25 #54D10 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2309.14632
openalex publication_date 2023/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show, in a certain specific sense, that both the density and the cardinality of a Hausdorff space are related to the "degree" to which the space is nonregular. It was shown by Sapirovskii that d(X)≤πχ(X)c(X) for a regular space X and the author observed this holds if the space is only quasiregular. We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree nq(X), which is countable when X is quasiregular, and showing d(X)≤πχ(X)c(X)nq(X) for any Hausdorff space X. This demonstrates that the degree to which a space is nonquasiregular has a fundamental and direct connection to its density and, ultimately, its cardinality. Importantly, if X is Hausdorff then nq(X) is "small" in the sense that nq(X)≤ψc(X). This results in a unified proof of both Sapirovskii's density bound for regular spaces and Sun's bound πχ(X)c(X)ψc(X) for the cardinality of a Hausdorff space X. A consequence is an improved bound for the cardinality of a Hausdorff space.