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 $$|X|\leq 2^{c(X)^{\pi\chi(X)}}$$ 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 $$|X|\leq 2^{d(X)^{\pi\chi(X)}}, \ |X|\leq d(X)^{\pi\chi(X)^{ot(X)}}, \text { and } |X|\leq 2^{\pi w(X)^{dot(X)}}, $$ 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 Šapirovskiĭ 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׳skiĭ-Šapirovskiĭ 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.