# Is every uniformizable topology induced by a Heine-Borel uniformity?

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.

• It's probably false, but one would need some more theory on the classs of uniform spaces with the HB property to disprove it, I think. Is this class closed under products, uniformly continuous images etc. ? – Henno Brandsma Dec 22 '18 at 7:59
• @HennoBrandsma It may be even easier; the proofs that topologies induced by Heine-Borel metrics are locally compact and separable may be able to be modified to show that topologies induced by Heine-Borel uniformities are locally compact and separable. – Keshav Srinivasan Dec 22 '18 at 8:19
• I don't think that would hold (separability and local compactness), btu I'm not sure. Just a hunch. – Henno Brandsma Dec 22 '18 at 10:25

A space $$X$$ is pseudocompact if any continuous $$X\to\mathbb R$$ is bounded. There exist completely regular non-compact pseudocompact spaces $$X$$ such as the long line (there are other examples at π-Base). By pseudocompactness, $$X$$ is bounded in any continuous pseudometric. By the positive answer to your previous question Is a set bounded in every metric for a uniformity bounded in the uniformity?, noting that the answers also work for pseudometrics in non-metrisable spaces, $$X$$ is bounded in every uniformity. So $$X$$ does not admit any Heine-Borel uniformity.