Some topological properties weaker than Lindelöfness

Abstract

A space $X$ is $C$-Lindel{ö}f (weakly $C$-Lindel{ö}f) if for everyclosed subset $F$ of $X$ and every open cover $\cal U$ of $F$ byopen subsets of $X$, there exists a countable subfamily $\cal V$ of$\cal U$ such that $F\subseteq \cup\{\overline V:V ın\calV\}$ (respectively, $F\subseteq \overline{\cup \cal V}$). In thispaper, we investigate the relationships among $C$-Lindel{ö}fspaces, weakly $C$-Lindel{ö}f spaces and Lindel{ö}f spaces, andalso study various properties of weakly $C$-Lindel{ö}f spaces and $C$-Lindel{ö}f spaces.