Definition 64.25.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. We say $X$ is regular at $x$ if $\mathcal{O}_{U, u}$ is a regular local ring for any (equivalently some) pair $(a : U \to X, u)$ consisting of an étale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$.

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