Show that in a compact Hausdorff space, \(A\) is closed iff \(A\) is compact.
Solution
1. (\(\Rightarrow\)) If \(A\) is closed, then \(A\) is compact (a closed subset of a compact space is compact).
closed subsets of compact spaces are compact.
2. (\(\Leftarrow\)) If \(A\) is compact, then \(A\) is closed (a compact subset of a Hausdorff space is closed).
compact subsets of Hausdorff spaces are closed.
3. Hence \(A\) is closed iff \(A\) is compact.
1 and 2.
4. Q.E.D.
3.