Suppose that \(X\) is compact and \(Y\) is Hausdorff. Prove that every one-to-one, onto, continuous map \(f:X\to Y\) is a homeomorphism.
Solution
1. \(f\) maps closed subsets of \(X\) to closed subsets of \(Y\).
Let \(C\subseteq X\) be closed. A closed subset of the compact space \(X\) is compact, so \(C\) is compact. The image \(f(C)\) is compact because \(f\) is continuous, and a compact subset of the Hausdorff space \(Y\) is closed.
2. \(f^{-1}\colon Y\to X\) is continuous.
For a closed set \(C\subseteq X\), \((f^{-1})^{-1}(C)=f(C)\) because \(f\) is bijective, and \(f(C)\) is closed by step 1.
3. Q.E.D.
\(f\) is a continuous bijection whose inverse is continuous by step 2, so \(f\) is a homeomorphism.