By Heine-Borel theorem, a closed and bounded subset of the Euclidean space is compact. If we analyze the proof, the only characteristic of Euclidean space that we need is: every bounded subset is contained in a compact subset. Is there a special name this kind of sets?
marked as duplicate by Alex Becker♦, Nate Eldredge, Asaf Karagila, t.b., Zhen Lin Mar 31 '12 at 0:04
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This proves that HB is equivalent to the fact that every bounded subset is a subset of a compact set. Clearly, I am doing it on a metric space, only then this makes sense. As pointed out OP seems to understand this equivalence and hence is irrelevant.
A metric space $(X, d)$ is said to be Heine-Borel if any closed and bounded subset of it is compact.
So, we can as well call these Heine-Borel spaces.
In fact, another characterisation of Heine Borel spaces is that, bounded sets are also totally bounded.