Lemma 67.4.3 (03KK)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.4: Separation axioms
Lemma 67.4.3 (cite)

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Lemma 67.4.3. 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 65.5.8, because a closed immersion of schemes is an immersion and is quasi-compact. $\square$

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