Show that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
Solution
It suffices to show that \(f\) is a closed map, i.e. if \(U\subseteq X\) is closed then \(f(U)\subseteq Y\) is again closed.
Let \(U\in X\) be closed; since \(X\) is closed, \(U\) is compact
- Since closed subsets of compact spaces are compact.
Since \(f\) is continuous, \(f(U)\) is compact
- Since the continuous image of a compact set is compact.
Since \(Y\) is Hausdorff and \(f(U)\) is compact, \(f(U)\) is closed
- Since compact subsets of Hausdorff spaces are closed.