The Stacks project

Remark 65.12.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $T \subset |X|$ be a locally closed subset. Let $\partial T$ be the boundary of $T$ in the topological space $|X|$. In a formula

\[ \partial T = \overline{T} \setminus T. \]

Let $U \subset X$ be the open subspace of $X$ with $|U| = |X| \setminus \partial T$, see Properties of Spaces, Lemma 64.4.8. Let $Z$ be the reduced closed subspace of $U$ with $|Z| = T$ obtained by taking the reduced induced closed subspace structure, see Properties of Spaces, Definition 64.12.5. By construction $Z \to U$ is a closed immersion of algebraic spaces and $U \to X$ is an open immersion, hence $Z \to X$ is an immersion of algebraic spaces over $S$ (see Spaces, Lemma 63.12.2). Note that $Z$ is a reduced algebraic space and that $|Z| = T$ as subsets of $|X|$. We sometimes say $Z$ is the reduced induced subspace structure on $T$.


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