Star versions of Lindel\"of spaces

A space $X$ is said to be set star-Lindel\"{o}f (resp., set strongly star-Lindel\"{o}f) 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 $A \subseteq {\rm St}( \bigcup \mathcal{V}, \mathcal{U})$ (resp., $A \subseteq {\rm St}( F, \mathcal{U})$). The classes of set star-Lindel\"{o}f spaces and set strongly star-Lindel\"{o}f spaces lie between the class of Lindel\"{o}f spaces and the class of star-Lindel\"{o}f spaces. In this paper, we investigate the relationship among set star-Lindel\"{o}f spaces, set strongly star-Lindel\"{o}f spaces, and other related spaces by providing some suitable examples and study the topological properties of set star-Lindel\"{o}f and set strongly star-Lindel\"{o}f spaces.