Lemma 87.37.2. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Let $T \subset |X_{red}|$ be a closed subset. Then the functor
is a formal algebraic space.
Proof. The functor $X_{/T}$ is an fppf sheaf since if $\{ U_ i \to U\} $ is an fppf covering, then $\coprod |U_ i| \to |U|$ is surjective.
Choose a covering $\{ g_ i : X_ i \to X\} _{i \in I}$ as in Definition 87.11.1. The morphisms $X_ i \times _ X X_{/T} \to X_{/T}$ are étale (see Spaces, Lemma 65.5.5) and the map $\coprod X_ i \times _ X X_{/T} \to X_{/T}$ is a surjection of sheaves. Thus it suffices to prove that $X_{/T} \times _ X X_ i$ is an affine formal algebraic space. A $U$-valued point of $X_ i \times _ X X_{/T}$ is a morphism $U \to X_ i$ whose image is contained in the closed subset $|g_{i, red}|^{-1}(T) \subset |X_{i, red}|$. Thus this follows from Lemma 87.37.1. $\square$
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