Ronnie Levy, Mikhail Matveev
Weak extent in normal spaces

Comment.Math.Univ.Carolinae 46,3 (2005) 497-501.

Abstract:If $X$ is a space, then the {weak extent} $we(X)$ of $X$ is the cardinal $\min \{\alpha :$ If $\Cal U$ is an open cover of $X$, then there exists $A\subseteq X$ such that $|A| = \alpha $ and $St(A,\Cal U)=X\}$. In this note, we show that if $X$ is a normal space such that $|X| = \frak c$ and $we(X) = \omega $, then $X$ does not have a closed discrete subset of cardinality $\frak c$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $\frak c$, even if the condition that $we(X) = \omega $ is replaced by the stronger condition that $X$ is separable.

Keywords: extent, weak extent, separable, star-Lindel\"{o}f, normal
AMS Subject Classification: Primary 54A25, 54D40