Show that an injective continuous map from a compact space to a Hausdorff space is an embedding (a homeomorphism onto its image).
Solution
Let \(f:X\to Y\) be continuous and injective, with \(X\) compact and \(Y\) Hausdorff.
1. The corestriction\[ f:X\longrightarrow f(X) \]is a continuous bijection.
2. It is a closed map.
If \(C\subseteq X\) is closed, then \(C\) is compact. Hence \(f(C)\) is compact in \(Y\), and compact subsets of a Hausdorff space are closed. Therefore \(f(C)\) is closed in the subspace \(f(X)\).
3. A bijective closed map has continuous inverse. Hence \(f:X\to f(X)\) is a homeomorphism, so \(f:X\to Y\) is an embedding.