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A bound for the density of any Hausdorff space

Nathan Carlson ⟨ncarlson@callutheran.edu⟩

Abstract:

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 speaker 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)minψc(X),L(X),pct(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. We give an example of a compact, Hausdorff space for which this new bound is a strict improvement over Sun’s bound.

Status: Accepted

Collection: Set-Theoretic Topology

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