Definition 66.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\Delta _{X/Y} : X \to X \times _ Y X$ be the diagonal morphism.

We say $f$ is

*separated*if $\Delta _{X/Y}$ is a closed immersion.We say $f$ is

*locally separated*^{1}if $\Delta _{X/Y}$ is an immersion.We say $f$ is

*quasi-separated*if $\Delta _{X/Y}$ is quasi-compact.

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