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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