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This is a follow-up to my question here. A subset of a topological space is called relatively compact if its closure is compact. My question is, what kind of topological spaces satisfy the following property: there exist countably many relatively compact sets $S_1,S_2,...$ such that every relatively compact set $S$ is a subset of some $S_n$? Or to put it another way, the collection of relatively compact sets has a countable cofinal subcollection.

Is there some category of topological spaces which satisfies this property? Maybe sigma-compact spaces?

My reason for asking this question, by the way, is that relatively compact sets form a bornology for $T_1$ spaces, and this property is one of the conditions for a bornology to be induced by a compatible metric, as I discuss here.

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I think hemicompactness comes close: this means that there are countably many compact $H_n\subseteq X$, $n \in \omega$, such that every compact subset of $X$ is a subset of some $H_n$. This condition is often considered when studying the compact-open topology on $X$. This property implies $\sigma$-compactness but is stronger: $\mathbb{Q}$ is $\sigma$-compact but not hemicompact (see this planetmath page), but for locally compact Hausdorff spaces the notions are equivalent.

It is clear that a hemicompact $X$ satisfies your condition, and a space satisfying your condition is hemicompact (using the closures of the base sets).

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