The Stacks project

Lemma 81.9.2. In Situation 81.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $\{ T_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $|Y|$. Then $\{ |f|^{-1}(T_ i)\} _{i \in I}$ is a locally finite collection of closed subsets of $X$.

Proof. Let $U \subset |X|$ be a quasi-compact open subset. Since the image $|f|(U) \subset |Y|$ is a quasi-compact subset there exists a quasi-compact open $V \subset |Y|$ such that $|f|(U) \subset V$. Note that

\[ \{ i \in I : |f|^{-1}(T_ i) \cap U \not= \emptyset \} \subset \{ i \in I : T_ i \cap V \not= \emptyset \} . \]

Since the right hand side is finite by assumption we win. $\square$

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