Compact-to-Hausdorff continuous bijections are homeomorphisms.Suggest a slogan
Let \(f\colon X\to Y\) be a continuous bijection. If \(X\) is compact and \(Y\) is Hausdorff, then \(f\) is a homeomorphism [@Mun00].
Closed subspaces of compact spaces are compact, continuous images of compact spaces are compact, and compact subspaces of Hausdorff spaces are closed [@Mun00]. Hence \(f\) is a closed map, so \(\inverseof{f}\) is continuous.
When is a continuous bijection from a compact space to a Hausdorff space a homeomorphism?