| Title:
|
A note on star Lindelöf, first countable and normal spaces (English) |
| Author:
|
Xuan, Wei-Feng |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
142 |
| Issue:
|
4 |
| Year:
|
2017 |
| Pages:
|
445-448 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A topological space $X$ is said to be star Lindelöf if for any open cover $\mathcal U$ of $X$ there is a Lindelöf subspace $A \subset X$ such that $\operatorname {St}(A, \mathcal U)=X$. The “extent” $e(X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$. We prove that under $V=L$ every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under $\rm MA +\nobreak \neg CH$, which shows that a star Lindelöf, first countable and normal space may not have countable extent. (English) |
| Keyword:
|
star Lindelöf space |
| Keyword:
|
first countable space |
| Keyword:
|
normal space |
| Keyword:
|
countable extent |
| MSC:
|
54D20 |
| MSC:
|
54E35 |
| idZBL:
|
Zbl 06819595 |
| idMR:
|
MR3739027 |
| DOI:
|
10.21136/MB.2017.0012-17 |
| . |
| Date available:
|
2017-11-20T15:03:57Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146980 |
| . |
| Reference:
|
[1] Bing, R. H.: Metrization of topological spaces.Can. J. Math. 3 (1951), 175-186. Zbl 0042.41301, MR 0043449, 10.4153/CJM-1951-022-3 |
| Reference:
|
[2] Engelking, R.: General Topology.Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989). Zbl 0684.54001, MR 1039321 |
| Reference:
|
[3] Fleissner, W. G.: Normal Moore spaces in the constructible universe.Proc. Am. Math. Soc. 46 (1974), 294-298. Zbl 0314.54028, MR 0362240, 10.2307/2039914 |
| Reference:
|
[4] Ginsburg, J., Woods, R. G.: A cardinal inequality for topological spaces involving closed discrete sets.Proc. Am. Math. Soc. 64 (1977), 357-360. Zbl 0398.54002, MR 0461407, 10.2307/2041457 |
| Reference:
|
[5] Hodel, R.: Cardinal functions I.Handbook of Set-Theoretic Topology K. Kunen et al. North-Holland, Amsterdam (1984), 1-61. Zbl 0559.54003, MR 0776620 |
| Reference:
|
[6] Miller, A. W.: Special subsets of the real line.Handbook of Set-Theoretic Topology K. Kunen et al. North-Holland, Amsterdam (1984), 201-233. Zbl 0588.54035, MR 0776624 |
| Reference:
|
[7] Tall, F. D.: Normality versus collectionwise normality.Handbook of Set-Theoretic Topology K. Kunen et al. North-Holland, Amsterdam (1984), 685-732. Zbl 0552.54011, MR 0776634 |
| Reference:
|
[8] Douwen, E. K. van, Reed, G. M., Roscoe, A. W., Tree, I. J.: Star covering properties.Topology Appl. 39 (1991), 71-103. Zbl 0743.54007, MR 1103993, 10.1016/0166-8641(91)90077-Y |
| Reference:
|
[9] Xuan, W. F., Shi, W. X.: Notes on star Lindelöf space.Topology Appl. 204 (2016), 63-69. Zbl 1342.54015, MR 3482703, 10.1016/j.topol.2016.02.009 |
| . |
