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I am reading a difinition of Lindelof $\Sigma$ space. It talked about compact cover. As the title explains, what does compact cover mean? It means every member of the cover is compact?

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It means that $\mathscr{C}$ is a cover of $X$ by compact sets.

Lindelöf $\Sigma$-spaces are $\Sigma$-spaces in the sense of Nagami that are also Lindelöf.

Definition. A space $X$ is a $\Sigma$-space in the sense of Nagami if there are a $\sigma$-locally finite collection $\mathscr{F}$ of closed subsets of $X$ and a cover $\mathscr{C}$ of $X$ by closed, countably compact sets such that if $C\in\mathscr{C}$, and $C\subseteq U$, where $U$ is an open set, then there is an $F\in\mathscr{F}$ such that $C\subseteq F\subseteq U$. (In other words, $\mathscr{F}$ is a network for the members of $\mathscr{C}$.)

If $X$ is Lindelöf, the closed, countably compact members of $\mathscr{C}$ are automatically compact.

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