The Stacks project

Lemma 5.23.5. Let $X$ be a spectral space. Let $E \subset X$ be a subset closed in the constructible topology (for example constructible).

  1. If $x \in \overline{E}$, then $x$ is the specialization of a point of $E$.

  2. If $E$ is stable under specialization, then $E$ is closed.

  3. If $E' \subset X$ is open in the constructible topology (for example constructible) and stable under generalization, then $E'$ is open.

Proof. Proof of (1). Let $x \in \overline{E}$. Let $\{ U_ i\} $ be the set of quasi-compact open neighbourhoods of $x$. A finite intersection of the $U_ i$ is another one. The intersection $U_ i \cap E$ is nonempty for all $i$. Since the subsets $U_ i \cap E$ are closed in the constructible topology we see that $\bigcap (U_ i \cap E)$ is nonempty by Lemma 5.23.2 and Lemma 5.12.6. Since $X$ is a sober space and $\{ U_ i\} $ is a fundamental system of open neighbourhoods of $x$, we see that $\bigcap U_ i$ is the set of generalizations of $x$. Thus $x$ is a specialization of a point of $E$.

Part (2) is immediate from (1).

Proof of (3). Assume $E'$ is as in (3). The complement of $E'$ is closed in the constructible topology (Lemma 5.15.2) and closed under specialization (Lemma 5.19.2). Hence the complement is closed by (2), i.e., $E'$ is open. $\square$

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