Star versions of Lindelöf spaces

Abstract

A space $X$ is said to be set star-Lindelöf (resp., set strongly star-Lindelöf) if for each nonempty subset $A$ of $X$ and each collection $\mathcal{U}$ of open sets in $X$ such that $\overline{A}\subseteq \bigcup \mathcal{U}$, there is a countable subset $\mathcal{V}$ of $\mathcal{U}$ (resp., countable subset $F$ of $\overline{A}$) such that $A\subseteq \mathrm{St}(\bigcup \mathcal{V},\mathcal{U})$ (resp., $A\subseteq \mathrm{St}(F,\mathcal{U})$). The classes of set star-Lindelöf spaces and set strongly star-Lindelöf spaces lie between the class of Lindelöf spaces and the class of star-Lindelöf spaces. In this paper, we investigate the relationship among set star-Lindelöf spaces, set strongly star-Lindelöf spaces, and other related spaces by providing some suitable examples and study the topological properties of set star-Lindelöf and set strongly star-Lindelöf spaces.

14 pages. arXiv admin note: text overlap with arXiv:2006.15380