Definition 64.13.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F$ be an algebraic space over $S$. Let $\Delta : F \to F \times F$ be the diagonal morphism.

We say $F$ is

*separated over $S$*if $\Delta $ is a closed immersion.We say $F$ is

*locally separated over $S$*^{1}if $\Delta $ is an immersion.We say $F$ is

*quasi-separated over $S$*if $\Delta $ is quasi-compact.We say $F$ is

*Zariski locally quasi-separated over $S$*^{2}if there exists a Zariski covering $F = \bigcup _{i \in I} F_ i$ such that each $F_ i$ is quasi-separated.

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