Proposition 5.16.4.label Let $X$ be a compact topological space, $Y$ be a Hausdorff space, and $f \in C(X; Y)$ be a bijection, then $f$ is a homeomorphism.
Proof. For each $K \subset X$ closed, $K$ and $f(K)$ are compact by Proposition 5.16.2. By Proposition 5.16.3, $f(K)$ is closed. Thus $f^{-1}$ maps closed sets to closed sets, and hence open sets to open sets.$\square$
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