New bounds on the cardinality of Hausdorff spaces and regular spaces

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 $|X|\le 2^{c(X{)}^{\pi \chi (X)}}$ and $|X|\le 2^{c(X)\pi \chi (X{)}^{ot(X)}}$, where the cardinal function $ot(X)$, introduced by Tkachenko, has the property $ot(X)\le \min \{t(X),c(X)\}$. 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 $|X|\le 2^{d(X{)}^{\pi \chi (X)}}$, $|X|\le d(X{)}^{\pi \chi (X{)}^{ot(X)}}$, and $|X|\le 2^{\pi w(X{)}^{dot(X)}}$, where $dot(X)\le \min \{ot(X),\pi \chi (X)\}$. 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 $w{\psi }_c(X)d{\psi }_c(X)\le {\psi }_c(X)$ for a Hausdorff space $X$, we show $|X|\le \pi \chi (X{)}^{c(X)w{\psi }_c(X)}$ if $X$ is regular and $|X|\le \pi \chi (X{)}^{c(X)d{\psi }_c(X)w{\psi }_c(X)}$ if $X$ is Hausdorff. This improves results of Sapirovskii and Sun. It is also shown that if $X$ is Hausdorff then $|X|\le 2^{d(X)w{\psi }_c(X)}$, 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.

19 pages