The Stacks project

Lemma 66.12.3. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Then $|i| : |Z| \to |X|$ is a homeomorphism onto a locally closed subset, and $i$ is a closed immersion if and only if the image $|i|(|Z|) \subset |X|$ is a closed subset.

Proof. The first statement is Properties of Spaces, Lemma 65.12.1. Let $U$ be a scheme and let $U \to X$ be a surjective ├ętale morphism. By assumption $T = U \times _ X Z$ is a scheme and the morphism $j : T \to U$ is an immersion of schemes. By Lemma 66.12.1 the morphism $i$ is a closed immersion if and only if $j$ is a closed immersion. By Schemes, Lemma 26.10.4 this is true if and only if $j(T)$ is closed in $U$. However, the subset $j(T) \subset U$ is the inverse image of $|i|(|Z|) \subset |X|$, see Properties of Spaces, Lemma 65.4.3. This finishes the proof. $\square$

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