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?