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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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