New bounds on the cardinality of Hausdorff spaces and regular spaces

TL;DR

This paper establishes new upper bounds on the cardinality of Hausdorff and regular spaces using weaker cardinal functions, improving previous results and providing insights into the structure of such spaces.

Contribution

The paper introduces novel bounds for the cardinality of Hausdorff and regular spaces that do not rely on the traditional $oldsymbol{ ext{psi}_c}$ function, enhancing existing theoretical frameworks.

Findings

New bounds for regular spaces: |X| ≤ 2^{c(X)^{πχ(X)}} and |X| ≤ 2^{c(X)πχ(X)^{ot(X)}}.

For Hausdorff spaces, bounds include |X| ≤ 2^{d(X)^{πχ(X)}} and |X| ≤ 2^{πw(X)^{dot(X)}}.

Improved cardinality bounds involving functions wψ_c(X) and dψ_c(X), surpassing previous results.

Abstract

Using weaker versions of the cardinal function $\psi_c(X)$, we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve $\psi_c(X)$ nor its variants at all. For example, we show if $X$ is regular then $|X|\leq 2^{c(X)^{\pi\chi(X)}}$ and $|X|\leq 2^{c(X)\pi\chi(X)^{ot(X)}}$, where the cardinal function $ot(X)$, introduced by Tkachenko, has the property $ot(X)\leq\min\{t(X),c(X)\}$. It follows from the latter that a regular space with cellularity at most $\mathfrak{c}$ and countable $\pi$-character has cardinality at most $2^\mathfrak{c}$. For a Hausdorff space $X$ we show $|X|\leq 2^{d(X)^{\pi\chi(X)}}$, $|X|\leq d(X)^{\pi\chi(X)^{ot(X)}}$, and $|X|\leq 2^{\pi w(X)^{dot(X)}}$, where $dot(X)\leq\min\{ot(X),\pi\chi(X)\}$. None of these bounds involve $\psi_c(X)$ or $\psi(X)$. By introducing the cardinal functions $w\psi_c(X)$ and $d\psi_c(X)$ with the property $w\psi_c(X)d\psi_c(X)\leq\psi_c(X)$ for a Hausdorff space $X$, we show $|X|\leq\pi\chi(X)^{c(X)w\psi_c(X)}$ if $X$ is regular and $|X|\leq\pi\chi(X)^{c(X)d\psi_c(X)w\psi_c(X)}$ if $X$ is Hausdorff. This improves results of Sapirovskii and Sun. It is also shown that if $X$ is Hausdorff then $|X|\leq 2^{d(X)w\psi_c(X)}$, which appears to be new even in the case where $w\psi_c(X)$ is replaced with $\psi_c(X)$. Compact examples show that $\psi(X)$ cannot be replaced with $d\psi_c(X)w\psi_c(X)$ in the bound $2^{\psi(X)}$ for the cardinality of a compact Hausdorff space $X$. Likewise, $\psi(X)$ cannot be replaced with $d\psi_c(X)w\psi_c(X)$ in the Arhangel'skii-Sapirovskii bound $2^{L(X)t(X)\psi(X)}$ for the cardinality of a Hausdorff space $X$. Finally, we make several observations concerning homogeneous spaces in this connection.