Lemma 42.13.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $\{ Z_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $Y$. Then $\{ f^{-1}(Z_ i)\} _{i \in I}$ 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
\[ \{ i \in I \mid f^{-1}(Z_ i) \cap U \not= \emptyset \} \subset \{ i \in I \mid Z_ i \cap V \not= \emptyset \} . \]
Since the right hand side is finite by assumption we win. $\square$
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