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

TL;DR

This paper investigates bounds on the cardinality of topological spaces with specific properties, proving new inequalities under set-theoretic assumptions and generalizing classical results like Arhangel'ski
2f's inequality.

Contribution

It establishes new upper bounds for the cardinality of various classes of topological spaces, generalizing classical inequalities and answering open questions under the Continuum Hypothesis.

Findings

Under CH, spaces with regular $G_\delta$-diagonals and caliber $\omega_1$ are separable.

Derived inequalities for the cardinality of Urysohn and Hausdorff spaces.

Generalized classical inequalities like Arhangel'ski
2f's bound.

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 $|X|\le 2^{\chi(X)\cdot L(X)}$.