Lemma 66.11.1. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $x \in |X|$. Consider étale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent

$x$ is a point of codimension $0$ on $X$,

for some $U \to X$ as above and $u \in U$ with $a(u) = x$, the point $u$ is the generic point of an irreducible component of $U$, and

for any $U \to X$ as above and any $u \in U$ mapping to $x$, the point $u$ is the generic point of an irreducible component of $U$.

If $X$ is representable, this is equivalent to $x$ being a generic point of an irreducible component of $|X|$.

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