# Is every separable locally compact metrizable topology induced by a Heine-Borel metric?

This is a follow-up to my question here. A metric has the Heine-Borel property if a set is closed and bounded with respect to the metric if and only if it is compact. Now if a metrizable topology is induced by a metric with the Heine-Borel property, then it is locally compact and separable.

My question is, is the converse true? That is, if a topologicy is separable, locally compact, and metrizable, then is it induced by some metric with the Heine-Borel property?

If not, what is an example of such a topology all of whose metrics fail to have the Heine-Borel property?

Such a locally compact separable metric space has a one-point compactification $$Y$$ that is metrisable. This is classical and well-known. Let $$d$$ be a metric for $$Y$$, consider $$X$$ to be a subset of $$Y$$ with $$Y\setminus X=\{\infty\}$$ and define $$d'(x_1, x_2) = d(x_1, x_2) + |\frac{1}{d(x_1,\infty)} - \frac{1}{d(x_2,\infty)}|$$ for $$x_1,x_2 \in X$$.
One can check that $$d$$ is a compatible metric on $$X$$ such that $$(X,d)$$ has the Heine-Borel property. Boundedness implies that we "stay away from $$\infty$$", essentially.