This is a follow-up to my question here. A subset $A$ of a uniform space is said to be bounded if for each entourage $V$, $A$ is a subset of $V^n[F]$ for some natural number $n$ and some finite set $F$. Just like for metric spaces, a compact subset of a uniform space is always closed and bounded, but the converse need not be true. A uniformity has the Heine-Borel property if a set is compact if and only if it is closed and bounded.
Suppose $X$ is a uniformizable topological space, AKA a completely regular topological space. My question is, is it necessarily the case that the topology on $X$ is induced by some Heine-Borel uniformity?
Now this answer shows that a metrizable topology is induced a metric with the Heine-Borel property if and only if it is locally compact and separable. But that doesn’t answer this question, because even if we take a topology which isn’t locally compact or isn’t separable, it’s possible that such a topology is induced by a non-metrizable uniformity with the Heine-Borel property.