Geodesic
A note on star Lindelöf, first countable and normal spaces
Mathematica Bohemica, Tome 142 (2017) no. 4, pp. 445-448
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/MB.2017.0012-17
Classification : 54D20, 54E35
Keywords: star Lindelöf space; first countable space; normal space; countable extent 
@article{10_21136_MB_2017_0012_17,
     author = {Xuan, Wei-Feng},
     title = {A note on star {Lindel\"of,} first countable and normal spaces},
     journal = {Mathematica Bohemica},
     pages = {445--448},
     year = {2017},
     volume = {142},
     number = {4},
     doi = {10.21136/MB.2017.0012-17},
     mrnumber = {3739027},
     zbl = {06819595},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0012-17/}
}
TY  - JOUR
AU  - Xuan, Wei-Feng
TI  - A note on star Lindelöf, first countable and normal spaces
JO  - Mathematica Bohemica
PY  - 2017
SP  - 445
EP  - 448
VL  - 142
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0012-17/
DO  - 10.21136/MB.2017.0012-17
LA  - en
ID  - 10_21136_MB_2017_0012_17
ER  - 
%0 Journal Article
%A Xuan, Wei-Feng
%T A note on star Lindelöf, first countable and normal spaces
%J Mathematica Bohemica
%D 2017
%P 445-448
%V 142
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2017.0012-17/
%R 10.21136/MB.2017.0012-17
%G en
%F 10_21136_MB_2017_0012_17
Xuan, Wei-Feng. A note on star Lindelöf, first countable and normal spaces. Mathematica Bohemica, Tome 142 (2017) no. 4, pp. 445-448. doi: 10.21136/MB.2017.0012-17

[1] Bing, R. H.: Metrization of topological spaces. Can. J. Math. 3 (1951), 175-186. | DOI | MR | JFM

[2] Engelking, R.: General Topology. Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989). | MR | JFM

[3] Fleissner, W. G.: Normal Moore spaces in the constructible universe. Proc. Am. Math. Soc. 46 (1974), 294-298. | DOI | MR | JFM

[4] Ginsburg, J., Woods, R. G.: A cardinal inequality for topological spaces involving closed discrete sets. Proc. Am. Math. Soc. 64 (1977), 357-360. | DOI | MR | JFM

[5] Hodel, R.: Cardinal functions I. Handbook of Set-Theoretic Topology K. Kunen et al. North-Holland, Amsterdam (1984), 1-61. | MR | JFM

[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. | MR | JFM

[7] Tall, F. D.: Normality versus collectionwise normality. Handbook of Set-Theoretic Topology K. Kunen et al. North-Holland, Amsterdam (1984), 685-732. | MR | JFM

[8] Douwen, E. K. van, Reed, G. M., Roscoe, A. W., Tree, I. J.: Star covering properties. Topology Appl. 39 (1991), 71-103. | DOI | MR | JFM

[9] Xuan, W. F., Shi, W. X.: Notes on star Lindelöf space. Topology Appl. 204 (2016), 63-69. | DOI | MR | JFM

Cité par Sources :