Regular \(G_\delta\)-diagonals and some upper bounds for cardinality of topological spaces

Abstract

We prove that, under CH, any space with a regular \(G_\delta\)-diagonal and caliber \(\omega_1\) is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space \(X\), we establish the inequality \(|X|\le wL(X)^{s\Delta_2(X)\cdot{dot(X)}}\) which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if \(X\) is a Hausdorff space, then \(|X|\le(\pi\chi(X)\cdot d(X))^{ot(X)\cdot\psi_c(X)}\); this result implies \v{S}apirovski{\u\i}'s inequality \(|X|\le\pi\chi(X)^{c(X)\cdot\psi(X)}\) which only holds for regular spaces. It is also proved that \(|X|\le \pi\chi(X)^{ot(X)\cdot\psi_c(X)\cdot aL_c(X)}\) for any Hausdorff space \(X\); this gives one more generalization of the famous Arhangel\(^\prime\)skii's inequality \(|X|\le 2^{\chi(X)\cdot L(X)}\).