This theorem is stated for only subsets of Rn. I was wondering if this theorem is true for subsets of general metric space. And what about general topological spaces?
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For the second question, this is a topological property known as pseudocompactness. – Apr 12 '21 at 08:23
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1Yes, true for all metric spaces, not for all topological spaces. A space with this property is in general called pseudocompact and there are locally compact Hausdorff pseudocompact but non-compact spaces. – Henno Brandsma Apr 12 '21 at 08:23