Lemma 66.24.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent

for any elementary étale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ has a unique minimal prime,

for any elementary étale neighbourhood $(U, u) \to (X, x)$ there is a unique irreducible component of $U$ through $u$,

for any elementary étale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ is unibranch,

the henselian local ring $\mathcal{O}_{X, x}^ h$ has a unique minimal prime.

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