Mathematica Bohemica, online first, 4 pp.


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
Classification MSC:  54D20, 54E35
DOI:  10.21136/MB.2017.0012-17

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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

 
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