The Stacks project

Proof. Let $X'$ and $Y'$ denote $X$ and $Y$ endowed with the constructible topology which are quasi-compact Hausdorff spaces by Lemma 5.23.2. Part (1) says $X' \to Y'$ is continuous and follows immediately from the definitions. Part (3) follows as $f(X')$ is a quasi-compact subset of the Hausdorff space $Y'$, see Lemma 5.12.4. We have a commutative diagram

of continuous maps of topological spaces. Since $Y'$ is Hausdorff we see that the fibres $X'_ y$ are closed in $X'$. As $X'$ is quasi-compact we see that $X'_ y$ is quasi-compact (Lemma 5.12.3). As $X'_ y \to X_ y$ is a surjective continuous map we conclude that $X_ y$ is quasi-compact (Lemma 5.12.7). $\square$