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The Stacks project

Lemma 5.24.8. Let X be a spectral space. Let E \subset X be a constructible subset. Let W \subset X be the set of points of X which specialize to a point of E. Then W \setminus E is a spectral space. If W = \bigcap U_ i with U_ i as in Lemma 5.24.7 (5) then W \setminus E = \mathop{\mathrm{lim}}\nolimits (U_ i \setminus E).

Proof. Since E is constructible, it is quasi-compact and hence Lemma 5.24.7 applies to W. If E is constructible, then E is constructible in U_ i for all i \in I. Hence U_ i \setminus E is spectral by Lemma 5.23.5. Since W \setminus E = \bigcap (U_ i \setminus E) we have W \setminus E = \mathop{\mathrm{lim}}\nolimits U_ i \setminus E by the universal property of limits. Then W \setminus E is a spectral space by Lemma 5.24.5. \square


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