The Stacks project

Lemma 17.19.4. Let $X$ be a spectral topological space. Let $\mathcal{B}$ be the set of quasi-compact open subsets of $X$. Let $\mathcal{F}$ be a sheaf of sets as in Equation ( Then there exist finitely many constructible closed subsets $Z_1, \ldots , Z_ n \subset X$ and finite sets $S_ i$ such that $\mathcal{F}$ is isomorphic to a subsheaf of $\prod (Z_ i \to X)_*\underline{S_ i}$.

Proof. By Lemma 17.19.3 we reduce to the case of a finite sober topological space and a sheaf with finite stalks. In this case $\mathcal{F} \subset \prod _{x \in X} i_{x, *}\mathcal{F}_ x$ where $i_ x : \{ x\} \to X$ is the embedding. We omit the proof that $i_{x, *}\mathcal{F}_ x$ is a constant sheaf on $\overline{\{ x\} }$. $\square$

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