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.