The Stacks project

Lemma 5.15.3. Let $f : X \to Y$ be a continuous map of topological spaces. If the inverse image of every retrocompact open subset of $Y$ is retrocompact in $X$, then inverse images of constructible sets are constructible.

Proof. This is true because $f^{-1}(U \cap V^ c) = f^{-1}(U) \cap f^{-1}(V)^ c$, combined with the definition of constructible sets. $\square$

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