The Stacks project

Lemma 5.15.14. Let $X$ be a topological space which has a basis for the topology consisting of quasi-compact opens. Let $E \subset X$ be a subset. Let $X = E_1 \cup \ldots \cup E_ m$ be a finite covering by constructible subsets. Then $E$ is constructible in $X$ if and only if $E \cap E_ j$ is constructible in $E_ j$ for each $j = 1, \ldots , m$.

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