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. This follows immediately from Lemma 5.17.7 which tells us that $f$ is closed, i.e., $f^{-1}$ is continuous. $\square$

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