The Stacks project

Lemma 75.7.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $T_ i \subset |X|$, $i = 1, \ldots , n$ be closed subsets. If $T_ i$, $i = 1, \ldots , n$ are proper over $Y$, then the same is true for $T_1 \cup \ldots \cup T_ n$.

Proof. Let $Z_ i$ be the reduced induced closed subscheme structure on $T_ i$. The morphism

\[ Z_1 \amalg \ldots \amalg Z_ n \longrightarrow X \]

is finite by Morphisms of Spaces, Lemmas 67.45.10 and 67.45.11. As finite morphisms are universally closed (Morphisms of Spaces, Lemma 67.45.9) and since $Z_1 \amalg \ldots \amalg Z_ n$ is proper over $S$ we conclude by Lemma 75.7.5 part (2) that the image $Z_1 \cup \ldots \cup Z_ n$ is proper over $S$. $\square$

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