The Stacks project

Lemma 5.15.6. Let $X$ be a topological space. Let $E \subset X$ be a subset. Let $X = V_1 \cup \ldots \cup V_ m$ be a finite covering by retrocompact opens. Then $E$ is constructible in $X$ if and only if $E \cap V_ j$ is constructible in $V_ j$ for each $j = 1, \ldots , m$.

Proof. If $E$ is constructible in $X$, then by Lemma 5.15.4 we see that $E \cap V_ j$ is constructible in $V_ j$ for all $j$. Conversely, suppose that $E \cap V_ j$ is constructible in $V_ j$ for each $j = 1, \ldots , m$. Then $E = \bigcup E \cap V_ j$ is a finite union of constructible sets by Lemma 5.15.5 and hence constructible. $\square$

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