Lemma 27.2.2. Let $X$ be a scheme and let $E \subset X$ be a locally constructible subset. Let $\xi \in X$ be a generic point of an irreducible component of $X$.

1. If $\xi \in E$, then an open neighbourhood of $\xi$ is contained in $E$.

2. If $\xi \not\in E$, then an open neighbourhood of $\xi$ is disjoint from $E$.

Proof. As the complement of a locally constructible subset is locally constructible it suffices to show (2). We may assume $X$ is affine and hence $E$ constructible (Lemma 27.2.1). In this case $X$ is a spectral space (Algebra, Lemma 10.25.2). Then $\xi \not\in E$ implies $\xi \not\in \overline{E}$ by Topology, Lemma 5.23.5 and the fact that there are no points of $X$ different from $\xi$ which specialize to $\xi$. $\square$

Comment #3440 by on

The statement says $E$ is constructible, but the proof seems to work even if $E$ is locally constructible. Maybe the statement should say $E$ is locally constructible, instead?

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