The Stacks project

Definition 66.12.5. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $Z \subset |X|$ be a closed subset. An algebraic space structure on $Z$ is given by a closed subspace $Z'$ of $X$ with $|Z'|$ equal to $Z$. The reduced induced algebraic space structure on $Z$ is the one constructed in Lemma 66.12.3. The reduction $X_{red}$ of $X$ is the reduced induced algebraic space structure on $|X|$.