Weak extent in normal spaces

Levy, Ronnie ; Matveev, Mikhail

Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 3, pp. 497-501.

Résumé

If $X$ is a space, then the {\it weak extent\/} $\operatorname{we}(X)$ 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 $\operatorname{St}(A,\Cal U)=X\}$. In this note, we show that if $X$ is a normal space such that $|X| = \frak c$ and $\operatorname{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 $\operatorname{we}(X) = \omega $ is replaced by the stronger condition that $X$ is separable.

Détail

Classification : 54A25, 54D15, 54D40
Keywords: extent; weak extent; separable; star-Lindel"{o}f; normal