The Stacks project

Lemma 5.15.4. Let $U \subset X$ be open. For a constructible set $E \subset X$ the intersection $E \cap U$ is constructible in $U$.

Proof. Suppose that $V \subset X$ is retrocompact open in $X$. It suffices to show that $V \cap U$ is retrocompact in $U$ by Lemma 5.15.3. To show this let $W \subset U$ be open and quasi-compact. Then $W$ is open and quasi-compact in $X$. Hence $V \cap W = V \cap U \cap W$ is quasi-compact as $V$ is retrocompact in $X$. $\square$

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