Compact space

Compact subsets of a Hausdorff space are closed

Let be a Hausdorff space. If is compact, then is closed topology

Proof

Let . For each assign an open neighbourhood of and likewise an open neighbourhood of such that for all (the Hausdorff property). Then is an open cover of and thus has a finite subcover . Then is an open subset of . Since every has an open neighbourhood , is open, whence is closed.

Similarly, Closed subsets of a compact space are compact.

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