Weak extent in normal spaces

Levy, Ronnie, and Matveev, Mikhail. "Weak extent in normal spaces." Commentationes Mathematicae Universitatis Carolinae 46.3 (2005): 497-501. <http://eudml.org/doc/249549>.

@article{Levy2005,
abstract = {If $X$ is a space, then the weak extent$\operatorname\{we\}(X)$ of $X$ is the cardinal $\min \lbrace \alpha :$ If $\mathcal \{U\}$ is an open cover of $X$, then there exists $A\subseteq X$ such that $|A| = \alpha $ and $\operatorname\{St\}(A,\mathcal \{U\})=X\rbrace $. In this note, we show that if $X$ is a normal space such that $|X| = \mathfrak \{c\}$ and $\operatorname\{we\}(X) = \omega $, then $X$ does not have a closed discrete subset of cardinality $\mathfrak \{c\}$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $\mathfrak \{c\}$, even if the condition that $\operatorname\{we\}(X) = \omega $ is replaced by the stronger condition that $X$ is separable.},
author = {Levy, Ronnie, Matveev, Mikhail},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {extent; weak extent; separable; star-Lindel"\{o\}f; normal; separable; star-Lindelöf; normal},
language = {eng},
number = {3},
pages = {497-501},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Weak extent in normal spaces},
url = {http://eudml.org/doc/249549},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Levy, Ronnie
AU - Matveev, Mikhail
TI - Weak extent in normal spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 3
SP - 497
EP - 501
AB - If $X$ is a space, then the weak extent$\operatorname{we}(X)$ of $X$ is the cardinal $\min \lbrace \alpha :$ If $\mathcal {U}$ is an open cover of $X$, then there exists $A\subseteq X$ such that $|A| = \alpha $ and $\operatorname{St}(A,\mathcal {U})=X\rbrace $. In this note, we show that if $X$ is a normal space such that $|X| = \mathfrak {c}$ and $\operatorname{we}(X) = \omega $, then $X$ does not have a closed discrete subset of cardinality $\mathfrak {c}$. We show that this result cannot be strengthened in ZFC to get that the extent of $X$ is smaller than $\mathfrak {c}$, even if the condition that $\operatorname{we}(X) = \omega $ is replaced by the stronger condition that $X$ is separable.
LA - eng
KW - extent; weak extent; separable; star-Lindel"{o}f; normal; separable; star-Lindelöf; normal
UR - http://eudml.org/doc/249549
ER -