Why the compactness of a metric/normed space and of a topology is so important as property ? I mean I understand the importance of completeness density and separability but I don’t see the point of the importance of compactness . Thanks
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Continuous functions on bounded intervals attain their extrema which is nice. Also Bolzano-Weierstrass, Arzelà-Ascoli and Stone-Weierstrass work only on compact sets. – bsbb4 Oct 30 '18 at 11:21
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1You may also want to read this, which also links to another similar question on math overflow. – user505379 Oct 30 '18 at 11:32