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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