A map from a compact space to a Hausdorff space is a proper.

Lemma 5.17.7. Let $f : X \to Y$ be a continuous map of topological spaces. If $X$ is quasi-compact and $Y$ is Hausdorff, then $f$ is proper.

Proof. Since every point of $Y$ is closed, we see from Lemma 5.12.3 that the closed subset $f^{-1}(y)$ of $X$ is quasi-compact for all $y \in Y$. Thus, by Theorem 5.17.5 it suffices to show that $f$ is closed. If $E \subset X$ is closed, then it is quasi-compact (Lemma 5.12.3), hence $f(E) \subset Y$ is quasi-compact (Lemma 5.12.7), hence $f(E)$ is closed in $Y$ (Lemma 5.12.4). $\square$

Comment #856 by Bhargav Bhatt on

Suggested slogan: A map from a compact space to a Hausdorff space is a proper.

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