If \(f: X_1 \to X_2\) is a homeomorphism of locally compact Hausdorff spaces, show \(f\) extends to a homeomorphism of their one-point compactifications.
Solution
Let \(X_i^*=X_i\cup\{\infty_i\}\) be the one-point compactifications. Define \[ f^*:X_1^*\to X_2^*,\qquad f^*(x)=f(x)\ (x\in X_1),\quad f^*(\infty_1)=\infty_2. \] This is bijective, with inverse extending \(f^{-1}\).
Continuity at ordinary points follows from continuity of \(f\). Let \(U\) be a neighborhood of \(\infty_2\). Then \[ X_2^*\setminus U=K \] is a compact subset of \(X_2\). Since \(f^{-1}(K)\) is compact in \(X_1\), \[ (f^*)^{-1}(U)=X_1^*\setminus f^{-1}(K) \] is a neighborhood of \(\infty_1\). Hence \(f^*\) is continuous at \(\infty_1\). The same argument applies to the inverse, so \(f^*\) is a homeomorphism.