A note on star Lindelöf, first countable and normal spaces

Xuan, Wei-Feng

Mathematica Bohemica, Tome 142 (2017) no. 4, pp. 445-448.

Résumé

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 $\mathrm{St}(A,\mathcal{U})=X$. 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 $\mathrm{M}\mathrm{A}+\text{nobreak}\mathrm{\neg }\mathrm{C}\mathrm{H}$, which shows that a star Lindelöf, first countable and normal space may not have countable extent.

Détail

DOI : 10.21136/MB.2017.0012-17

Classification : 54D20, 54E35
Mots-clés : star Lindelöf space; first countable space; normal space; countable extent