On cardinality bounds involving the weak Lindel\"of degree

Abstract

We give a general closing-off argument in Theorem 2.1 from which several corollaries follow, including (1) if $X$ is a locally compact Hausdorff space then $|X|\leq 2^{wL(X)\psi(X)}$, and (2) if $X$ is a locally compact power homogeneous Hausdorff space then $|X|\leq 2^{wL(X)t(X)}$. The first extends the well-known cardinality bound $2^{\psi(X)}$ for a compactum $X$ in a new direction. As $|X|\leq 2^{wL(X)\chi(X)}$ for a normal space $X$ [3], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.10 we give a short, direct proof of (1) that does not use 2.1. Yet 2.1 is broad enough to establish results much more general than (1), such as if $X$ is a regular space with a $\pi$-base $\scr{B}$ such that $|B|\leq 2^{wL(X)\chi(X)}$ for all $B\in\scr{B}$, then $|X|\leq 2^{wL(X)\chi(X)}$. Separately, it is shown that if $X$ is a regular space with a $\pi$-base whose elements have compact closure, then $|X|\leq 2^{wL(X)\psi(X)t(X)}$. This partially answers a question from [3] and gives a third, separate proof of (1). We also show that if $X$ is a weakly Lindel\"of, normal, sequential space with $\chi(X)\leq 2^{\aleph_0}$, then $|X|\leq 2^{\aleph_0}$. Result (2) above is a new generalization of the cardinality bound $2^{t(X)}$ for a power homogeneous compactum $X$ (Arhangel'skii, van Mill, and Ridderbos [2], De la Vega in the homogeneous case [9]). To this end we show that if $U\subseteq clD\subseteq X$, where $X$ is power homogeneous and $U$ is open, then $|U|\leq |D|^{\pi_{\chi}(X)}$. This is a strengthening of a result of Ridderbos [18].