The Stacks project

Lemma 27.2.5. Let $X$ be a quasi-compact and quasi-separated scheme. Any locally constructible subset of $X$ is constructible.

Proof. As $X$ is quasi-compact we can choose a finite affine open covering $X = V_1 \cup \ldots \cup V_ m$. As $X$ is quasi-separated each $V_ i$ is retrocompact in $X$ by Lemma 27.2.3. Hence by Topology, Lemma 5.15.6 we see that $E \subset X$ is constructible in $X$ if and only if $E \cap V_ j$ is constructible in $V_ j$. Thus we win by Lemma 27.2.1. $\square$


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