Star versions of Lindel\"of spaces

TL;DR

This paper introduces and explores set star-Lindel"of and set strongly star-Lindel"of spaces, analyzing their properties and relationships with other topological spaces through examples and theoretical study.

Contribution

It defines new classes of spaces, investigates their properties, and clarifies their position between Lindel"of and star-Lindel"of spaces.

Findings

Set star-Lindel"of spaces are between Lindel"of and star-Lindel"of spaces.

Set strongly star-Lindel"of spaces are a related class with distinct properties.

The paper provides examples illustrating the relationships among these classes.

Abstract

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.