The Stacks project

Lemma 66.19.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Let $(U, \overline{u})$ an étale neighborhood of $\overline{x}$. Let $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ be an étale covering in $X_{spaces, {\acute{e}tale}}$. Then there exist $i \in I$ and $\overline{u}_ i : \overline{x} \to U_ i$ such that $\varphi _ i : (U_ i, \overline{u}_ i) \to (U, \overline{u})$ is a morphism of étale neighborhoods.

Proof. Let $u \in |U|$ be the image of $\overline{u}$. As $|U| = \bigcup _{i \in I} \varphi _ i(|U_ i|)$ there exists an $i$ and a point $u_ i \in U_ i$ mapping to $x$. Apply Lemma 66.19.4 to $(U_ i, u_ i) \to (U, u)$ and $\overline{u}$ to get the desired geometric point. $\square$

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