What topological spaces satisfy a property involving relatively compact sets?

Question

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.

Answer 1

This is equivalent to local compactness.

If a topological space satisfies your property, then a singleton point set $\{x\}$ is compact, and hence there must be a relatively compact set $T$ whose interior contains $x$ (in fact, it contains $\overline{\{x\}}$, which may be strictly larger). The closure of $T$ is a compact neighbourhood of $x$.

If, on the other hand, if we have locally compact space and a relatively compact set $S$, then $\overline{S}$ is compact. For each point $s \in \overline{S}$, there is an open neighbourhood $\mathcal{N}_s$ of $s$ that is compact. Note that the interiors of these neighbourhoods cover $\overline{S}$, hence there must be a finite-subcover of $\overline{S}$. Unioning this finite subcover yields an open, relatively compact neighbourhood containing $\overline{S}$, establishing equivalence.