# A name for metric spaces where every closed and bounded subset is compact

On this page it says that a metric space (or topological vector space) is said to have the Heine–Borel property if every closed and bounded subset is compact. But today on Wikipedia it says that a metric space (or, for instance, a topological vector space) is said to have the Heine–Borel property if every of its open covers has a finite subcover, which is not equivalent.

I want to be able to talk about metric spaces where every closed and bounded subset is compact, but I now do not know how to refer to this property.

• If you google "Heine-Borel property," you'll see instances of both uses, so it's not a particularly standardized term. (Though it's a bit odd, given that the definition Wikipedia gives is precisely the definition of a compact set in a topological space - also, other references using the word in this sense seem to refer to Wikipedia, so this might really be an error that got propagated) – Milo Brandt Apr 26 '17 at 20:04