The Stacks project

Lemma 65.5.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is quasi-compact if and only if $|X|$ is quasi-compact.

Proof. Choose a scheme $U$ and an ├ętale surjective morphism $U \to X$. We will use Lemma 65.4.4. If $U$ is quasi-compact, then since $|U| \to |X|$ is surjective we conclude that $|X|$ is quasi-compact. If $|X|$ is quasi-compact, then since $|U| \to |X|$ is open we see that there exists a quasi-compact open $U' \subset U$ such that $|U'| \to |X|$ is surjective (and still ├ętale). Hence we win. $\square$

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