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: for every relatively compact set $S$, there exists a relatively compact set $T$ such that the closure of $S$ is a subset of the interior of $T$?
Is there some category of topological spaces which satisfies this property? And is there an example of a $T_1$ space which does not satisfy this property?
My reason for asking this, by the way, is that relatively compact subsets of a $T_1$ form a bornology, and this property says that that bornology interacts well with the topology.