Compact space

Closed subsets of a compact space are compact

Let be a compact space. Then if is closed it is also compact. #m/thm/topology

Proof

Let be an open cover of . Then is an open cover of , so by compactness it has a finite subcover . But it follows that is a finite subcover of . Hence is compact.

Similarly, Compact subsets of a Hausdorff space are closed.

#state/tidy | #lang/en | #SemBr