| Title:
|
Weak extent in normal spaces (English) |
| Author:
|
Levy, Ronnie |
| Author:
|
Matveev, Mikhail |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
497-501 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
extent |
| Keyword:
|
weak extent |
| Keyword:
|
separable |
| Keyword:
|
star-Lindel"{o}f |
| Keyword:
|
normal |
| MSC:
|
54A25 |
| MSC:
|
54D15 |
| MSC:
|
54D40 |
| idZBL:
|
Zbl 1121.54012 |
| idMR:
|
MR2174527 |
| . |
| Date available:
|
2009-05-05T16:52:45Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119543 |
| . |
| Reference:
|
[H] Hodel R.E.: Combinatorial set theory and cardinal function inequalities.Proc. Amer. Math. Soc. 111 (1991), 567-575. Zbl 0713.54007, MR 1039531 |
| Reference:
|
[I] Ikenaga S.: A class which contains Lindelöf spaces, separable spaces and countably compact spaces.Mem. Numazo Coll. Technology 18 (1983), 105-108. |
| Reference:
|
[K] Kozma G.: On removing one point from a compact space.Houston J. Math. 30 4 (2004), 1115-1126. Zbl 1069.54017, MR 2110253 |
| Reference:
|
[M] Matveev M.: How weak is weak extent?.Topology Appl. 119 (2002), 229-232. Zbl 0986.54003, MR 1886097 |
| Reference:
|
[T1] Tall F.: Normality versus collectionwise normality.Handbook of Set Theoretic Topology, North-Holland, Amsterdam, 1984, pp.685-732. Zbl 0552.54011, MR 0776634 |
| Reference:
|
[T2] Tall F.: Weakly collectionwise Hausdorff spaces.Topology Proc. 1 (1976), 295-304. Zbl 0382.54004, MR 0454914 |
| . |
