Show that a continuous map from a compact space to a Hausdorff space is closed.
Exercise
Solution
1. Let \(f:X\to Y\) be continuous, with \(X\) compact and \(Y\) Hausdorff, and let \(F\subseteq X\) be closed.
To show that \(f\) is closed, it suffices to prove that \(f(F)\) is closed for every closed \(F\).
2. The subset \(F\) is compact.
A closed subset of a compact space is compact.
3. The image \(f(F)\) is compact.
Continuous images of compact spaces are compact.
4. Since \(Y\) is Hausdorff, \(f(F)\) is closed in \(Y\).
Compact subsets of Hausdorff spaces are closed.
5. Therefore \(f\) is a closed map.
Apply 4 to every closed subset \(F\subseteq X\).