The Stacks project

Lemma 5.24.7. Let $W$ be a subset of a spectral space $X$. The following are equivalent:

  1. $W$ is an intersection of constructible sets and closed under generalizations,

  2. $W$ is quasi-compact and closed under generalizations,

  3. there exists a quasi-compact subset $E \subset X$ such that $W$ is the set of points specializing to $E$,

  4. $W$ is an intersection of quasi-compact open subsets,

  5. there exists a nonempty set $I$ and quasi-compact opens $U_ i \subset X$, $i \in I$ such that $W = \bigcap U_ i$ and for all $i, j \in I$ there exists a $k \in I$ with $U_ k \subset U_ i \cap U_ j$.

In this case we have (a) $W$ is a spectral space, (b) $W = \mathop{\mathrm{lim}}\nolimits U_ i$ as topological spaces, and (c) for any open $U$ containing $W$ there exists an $i$ with $U_ i \subset U$.

Proof. Let $W \subset X$ satisfy (1). Then $W$ is closed in the constructible topology, hence quasi-compact in the constructible topology (by Lemmas 5.23.2 and 5.12.3), hence quasi-compact in the topology of $X$ (because opens in $X$ are open in the constructible topology). Thus (2) holds.

It is clear that (2) implies (3) by taking $E = W$.

Let $X$ be a spectral space and let $E \subset W$ be as in (3). Since every point of $W$ specializes to a point of $E$ we see that an open of $W$ which contains $E$ is equal to $W$. Hence since $E$ is quasi-compact, so is $W$. If $x \in X$, $x \not\in W$, then $Z = \overline{\{ x\} }$ is disjoint from $W$. Since $W$ is quasi-compact we can find a quasi-compact open $U$ with $W \subset U$ and $U \cap Z = \emptyset $. We conclude that (4) holds.

If $W = \bigcap _{j \in J} U_ j$ then setting $I$ equal to the set of finite subsets of $J$ and $U_ i = U_{j_1} \cap \ldots \cap U_{j_ r}$ for $i = \{ j_1, \ldots , j_ r\} $ shows that (4) implies (5). It is immediate that (5) implies (1).

Let $I$ and $U_ i$ be as in (5). Since $W = \bigcap U_ i$ we have $W = \mathop{\mathrm{lim}}\nolimits U_ i$ by the universal property of limits. Then $W$ is a spectral space by Lemma 5.24.5. Let $U \subset X$ be an open neighbourhood of $W$. Then $E_ i = U_ i \cap (X \setminus U)$ is a family of constructible subsets of the spectral space $Z = X \setminus U$ with empty intersection. Using that the spectral topology on $Z$ is quasi-compact (Lemma 5.23.2) we conclude from Lemma 5.12.6 that $E_ i = \emptyset $ for some $i$. $\square$

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