The Stacks project

Lemma 64.11.2. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. The set of codimension $0$ points of $X$ is dense in $|X|$.

Proof. If $U$ is a scheme, then the set of generic points of irreducible components is dense in $U$ (holds for any quasi-sober topological space). Thus if $U \to X$ is a surjective ├ętale morphism, then the set of codimension $0$ points of $X$ is the image of a dense subset of $|U|$ (Lemma 64.11.1). Since $|X|$ has the quotient topology for $|U| \to |X|$ we conclude. $\square$


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