Lemma 5.15.8 (09YF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.15: Constructible sets
Lemma 5.15.8 (cite)

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Lemma 5.15.8. Let $X$ be a topological space. Let $T \subset X$ be a subset. Suppose

$T$ is retrocompact in $X$ ,
quasi-compact opens form a basis for the topology on $X$ .

Then for a constructible set $E \subset X$ the intersection $E \cap T$ is constructible in $T$ .

Proof. Suppose that $V \subset X$ is retrocompact open in $X$ . It suffices to show that $V \cap T$ is retrocompact in $T$ by Lemma 5.15.3. To show this let $W \subset T$ be open and quasi-compact. By assumption (2) we can find a quasi-compact open $W' \subset X$ such that $W = T \cap W'$ (details omitted). Hence $V \cap T \cap W = V \cap T \cap W'$ is the intersection of $T$ with the quasi-compact open $V \cap W'$ as $V$ is retrocompact in $X$ . Thus $V \cap T \cap W$ is quasi-compact. $\square$

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