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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