Lemma 5.23.4. Let $X$ and $Y$ be spectral spaces. Let $f : X \to Y$ be a continuous map. Then $f$ is spectral if and only if $f$ is continuous in the constructible topology.
Proof. The only if part of this is Lemma 5.23.3. Assume $f$ is continuous in the constructible topology. Let $V \subset Y$ be quasi-compact open. Then $V$ is open and closed in the constructible topology. Hence $f^{-1}(V)$ is open and closed in the constructible topology. Hence $f^{-1}(V)$ is quasi-compact in the constructible topology as $X$ is quasi-compact in the constructible topology by Lemma 5.23.2. Since the identity $f^{-1}(V) \to f^{-1}(V)$ is surjective and continuous from the constructible topology to the usual topology, we conclude that $f^{-1}(V)$ is quasi-compact in the topology of $X$ by Lemma 5.12.7. This finishes the proof. $\square$
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