The Stacks project

Lemma 82.7.5. In Situation 82.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is quasi-compact, and $\{ T_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $|X|$. Then $\{ \overline{|f|(T_ i)}\} _{i \in I}$ is a locally finite collection of closed subsets of $|Y|$.

Proof. Let $V \subset |Y|$ be a quasi-compact open subset. Then $|f|^{-1}(V) \subset |X|$ is quasi-compact by Morphisms of Spaces, Lemma 67.8.3. Hence the set $\{ i \in I : T_ i \cap |f|^{-1}(V) \not= \emptyset \} $ is finite by a simple topological argument which we omit. Since this is the same as the set

\[ \{ i \in I : |f|(T_ i) \cap V \not= \emptyset \} = \{ i \in I : \overline{|f|(T_ i)} \cap V \not= \emptyset \} \]

the lemma is proved. $\square$

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