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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.

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    $\begingroup$ 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) $\endgroup$ – Milo Brandt Apr 26 '17 at 20:04
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I would not hesitate to refer to this as the Heine-Borel property (though you will probably want to define the term anyways since it is not that widely known). In my experience, despite what Wikipedia says, the use of "Heine-Borel property" to mean just "compact" is quite rare, and is mostly only done in expositions of the basic theory of compactness in metric spaces (to contrast with other equivalent definitions of compactness like Bolzano-Weierstrass).

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If one wants standard terminology, such spaces are called proper, see this wikipedia article, as well as Borel compact, see the discussion here.

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