Lemma 5.15.7 (09YE)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.15: Constructible sets
Lemma 5.15.7 (cite)

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Lemma 5.15.7. Let $X$ be a topological space. Let $Z \subset X$ be a closed subset such that $X \setminus Z$ is quasi-compact. Then for a constructible set $E \subset X$ the intersection $E \cap Z$ is constructible in $Z$ .

Proof. Suppose that $V \subset X$ is retrocompact open in $X$ . It suffices to show that $V \cap Z$ is retrocompact in $Z$ by Lemma 5.15.3. To show this let $W \subset Z$ be open and quasi-compact. The subset $W' = W \cup (X \setminus Z)$ is quasi-compact, open, and $W = Z \cap W'$ . Hence $V \cap Z \cap W = V \cap Z \cap W'$ is a closed subset of the quasi-compact open $V \cap W'$ as $V$ is retrocompact in $X$ . Thus $V \cap Z \cap W$ is quasi-compact by Lemma 5.12.3. $\square$

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