Some topological properties weaker than Lindelofness

Yan-Kui Song

A space $X$ is $C$-Lindel{}f (weakly $C$-Lindel{}f) if for every closed subset $F$ of $X$ and every open cover $\cal U$ of $F$ by open subsets of $X$, there exists a countable subfamily $\cal V$ of $\cal U$ such that $F\subseteq \cup\{\overline V:V \in\cal V\}$ (respectively, $F\subseteq \overline{\cup \cal V}$). In this paper, we investigate the relationships among $C$-Lindel{}f spaces, weakly $C$-Lindel{}f spaces and Lindel{}f spaces, and also study various properties of weakly $C$-Lindel{}f spaces and $C$-Lindel{}f spaces.