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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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