Paper detail

New bounds on the cardinality of Hausdorff spaces and regular spaces

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