The Stacks project

Lemma 66.24.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space.

Proof. A quasi-separated algebraic space has an underlying sober topological space, see Lemma 66.15.1. It is Noetherian by Lemma 66.24.2. $\square$

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