Show that a continuous map from a compact space to a Hausdorff space is closed.
Solution
1. Let \(f : X \to Y\) be continuous, with \(X\) compact and \(Y\) Hausdorff, and let \(C \subseteq X\) be closed.
setup.
2. \(C\) is compact (a closed subset of a compact space is compact).
1.
3. \(f(C)\) is compact (the continuous image of a compact set is compact).
2 and continuity.
4. \(f(C)\) is closed (a compact subset of a Hausdorff space is closed).
3 and \(Y\) Hausdorff.
5. Hence \(f\) maps closed sets to closed sets, so \(f\) is a closed map.
1 and 4.
6. Q.E.D.
5.