More on Cardinality Bounds Involving the Weak Lindel\"of degree

TL;DR

This paper establishes new bounds on the cardinality of Hausdorff topological spaces involving the weak Lindel"of degree, extending known results for special classes like extremally disconnected and power homogeneous spaces.

Contribution

It introduces several new cardinality bounds for various classes of topological spaces using the weak Lindel"of degree and other invariants, improving previous results.

Findings

For extremally disconnected spaces, |X| ≤ 2^{wL(X)πχ(X)ψ(X)}.

For spaces with certain diagonal properties, |X| ≤ 2^{aleph_0}.

In locally compact and power homogeneous spaces, bounds involve wL(X) and other invariants.

Abstract

We give several new bounds for the cardinality of a Hausdorff topological space $X$ involving the weak Lindel\"of degree $wL(X)$. In particular, we show that if $X$ is extremally disconnected, then $|X|\leq 2^{wL(X)\pi\chi(X)\psi(X)}$, and if $X$ is additionally power homogeneous, then $|X|\leq 2^{wL(X)\pi\chi(X)}$. We also prove that if $X$ is an almost Lindel\"of space with a strong $G_\delta$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$; that if $X$ is a star-cdc space with a $G_\delta$-diagonal of rank 3, then $|X| \le 2^{\aleph_0}$; and if $X$ is any normal star-cdc space $X$ with a $G_\delta$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$. Several improvements of results in [9] are also given. We show that if $X$ is locally compact, then $|X|\leq wL(X)^{\psi(X)}$ and that $|X|\leq wL(X)^{t(X)}$ if $X$ is additionally power homogeneous. We also prove that $|X|\leq 2^{\psi_c(X)t(X)wL(X)}$ for any space with a $\pi$-base whose elements have compact closures and that the stronger inequality $|X|\leq wL(X)^{\psi_c(X)t(X)}$ is true when $X$ is locally $H$-closed or locally Lindel\"of.