Lemma 5.17.8. Let $f : X \to Y$ be a continuous map of topological spaces. If $f$ is bijective, $X$ is quasi-compact, and $Y$ is Hausdorff, then $f$ is a homeomorphism.

Proof. It suffices to prove $f$ is closed, because this implies that $f^{-1}$ is continuous. If $T \subset X$ is closed, then $T$ is quasi-compact by Lemma 5.12.3, hence $f(T)$ is quasi-compact by Lemma 5.12.7, hence $f(T)$ is closed by Lemma 5.12.4. $\square$

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