Lemma 42.11.2. 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 $f$ is quasi-compact, and $\{ Z_ i\} _{i \in I}$ is a locally finite collection of closed subsets of $X$. Then $\{ \overline{f(Z_ i)}\} _{i \in I}$ is a locally finite collection of closed subsets of $Y$.

**Proof.**
Let $V \subset Y$ be a quasi-compact open subset. Since $f$ is quasi-compact the open $f^{-1}(V)$ is quasi-compact. Hence the set $\{ i \in I \mid Z_ 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

the lemma is proved. $\square$

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## Comments (2)

Comment #123 by Adeel Ahmad Khan on

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