Lemma 65.4.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ is locally separated and $f$ is quasi-separated.
Proof. This is true, via the general principle Spaces, Lemma 63.5.8, because a closed immersion of schemes is an immersion and is quasi-compact. $\square$
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