# What topological spaces satisfy another property involving relatively compact sets?

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.

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).