The Stacks project

Lemma 5.19.10. Let $X$ be a Noetherian sober topological space. Let $E \subset X$ be a subset of $X$.

  1. If $E$ is constructible and stable under specialization, then $E$ is closed.

  2. If $E$ is constructible and stable under generalization, then $E$ is open.

Proof. Let $E$ be constructible and stable under generalization. Let $Y \subset X$ be an irreducible closed subset with generic point $\xi \in Y$. If $E \cap Y$ is nonempty, then it contains $\xi $ (by stability under generalization) and hence is dense in $Y$, hence it contains a nonempty open of $Y$, see Lemma 5.16.3. Thus $E$ is open by Lemma 5.16.5. This proves (2). To prove (1) apply (2) to the complement of $E$ in $X$. $\square$

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