A bound for the density of any Hausdorff space

TL;DR

This paper establishes a new bound relating the density and cardinality of Hausdorff spaces to their degree of nonquasiregularity, unifying and extending previous results for regular and quasiregular spaces.

Contribution

The paper introduces the nonquasiregularity degree $nq(X)$ for Hausdorff spaces and proves a generalized density bound, extending prior bounds for regular and quasiregular spaces.

Findings

Derived a bound: $d(X) \,\leq\, \pi\chi(X)^{c(X)nq(X)}$ for all Hausdorff spaces.

Showed $nq(X) \leq \psi_c(X)$ for Hausdorff spaces, linking nonquasiregularity to other topological invariants.

Unified proofs of existing bounds for regular and Hausdorff spaces, leading to an improved cardinality bound.

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 $d(X)\leq\pi\chi(X)^{c(X)}$ 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 $d(X)\leq\pi\chi(X)^{c(X)nq(X)}$ 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 $nq(X)\leq\psi_c(X)$. This results in a unified proof of both Sapirovskii's density bound for regular spaces and Sun's bound $\pi\chi(X)^{c(X)\psi_c(X)}$ for the cardinality of a Hausdorff space $X$. A consequence is an improved bound for the cardinality of a Hausdorff space.