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 {\rm St}( \bigcup \mathcal{V}, \mathcal{U})$ (resp., $ A \subseteq {\rm 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.