Show that an injective continuous map from a compact space to a Hausdorff space is an embedding (a homeomorphism onto its image).
Solution
Goal: Show that an injective continuous map \(f: X \to Y\) from a compact space \(X\) to a Hausdorff space \(Y\) is an embedding: a homeomorphism onto its image.
1. \(f: X \to f(X)\) is a continuous bijection.
\(f\) is injective by hypothesis, so viewed as a map onto its image it is bijective; continuity is inherited.
2. \(f(X)\) is Hausdorff.
Subspaces of Hausdorff spaces are Hausdorff, and \(Y\) is Hausdorff.
3. \(f\) maps closed sets to closed sets (in \(f(X)\)).
3.1. Let \(C \subseteq X\) be closed; then \(C\) is compact.
Closed subsets of compact spaces are compact.
3.2. \(f(C)\) is compact in \(Y\).
Continuous images of compact sets are compact.
3.3. \(f(C)\) is closed in \(Y\), hence closed in \(f(X)\).
Compact subsets of Hausdorff spaces are closed; and closed in \(Y\) implies closed in the subspace \(f(X)\).
4. \(f^{-1}: f(X) \to X\) is continuous.
A bijective map is a homeomorphism iff it is a closed map (equivalently: \(f^{-1}\) is continuous iff for every closed \(C \subseteq X\), \((f^{-1})^{-1}(C) = f(C)\) is closed in \(f(X)\)), and 3 gives closedness.
5. Q.E.D.
1 and 4 show \(f: X \to f(X)\) is a homeomorphism, i.e. \(f\) is an embedding.