Mathematica Bohemica, Vol. 142, No. 4, pp. 445-448, 2017
A note on star Lindelöf, first countable and normal spaces
Wei-Feng Xuan
Received January 30, 2017. First published May 2, 2017.
Abstract: 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 ${\rm MA + \neg CH}$, which shows that a star Lindelöf, first countable and normal space may not have countable extent.
Keywords: star Lindeöf space; first countable space; normal space; countable extent
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Affiliations: Wei-Feng Xuan, School of Science, Nanjing Audit University, 86 YuShan Road, Nanjing, China, 210093, e-mail: wfxuan@nau.edu.cn
