# Weak extent in normal spaces

Commentationes Mathematicae Universitatis Carolinae (2005)

- Volume: 46, Issue: 3, page 497-501
- ISSN: 0010-2628

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topLevy, 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 -

## References

top- Hodel R.E., Combinatorial set theory and cardinal function inequalities, Proc. Amer. Math. Soc. 111 (1991), 567-575. (1991) Zbl0713.54007MR1039531
- Ikenaga S., A class which contains Lindelöf spaces, separable spaces and countably compact spaces, Mem. Numazo Coll. Technology 18 (1983), 105-108. (1983)
- Kozma G., On removing one point from a compact space, Houston J. Math. 30 4 (2004), 1115-1126. (2004) Zbl1069.54017MR2110253
- Matveev M., How weak is weak extent?, Topology Appl. 119 (2002), 229-232. (2002) Zbl0986.54003MR1886097
- Tall F., Normality versus collectionwise normality, Handbook of Set Theoretic Topology, North-Holland, Amsterdam, 1984, pp.685-732. Zbl0552.54011MR0776634
- Tall F., Weakly collectionwise Hausdorff spaces, Topology Proc. 1 (1976), 295-304. (1976) Zbl0382.54004MR0454914

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