So the definition I have seen is:(adsbygoogle = window.adsbygoogle || []).push({});

Given a topological space <S,F> it is compact if for any cover (union of open sets which is equal to S) there exists a finite subcover.

By the definition of a topological space both S and the empty set must belong to the family of subsets F.

Wouldn't <S, empty set> be a finite subcover for S? In which case S is compact.

By this sort of logic any open subset X of a topological space S is also compact in the relative topology since X will belong to the family of subsets in the relative topology so <X, empty set> would be a finite subcover for X.

I'm assuming the resolution to this is that the finite subcover cannot include the space itself but I just want to double check I haven't horribly misunderstood something.

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# Confusion about definition of compactness

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