Definition 66.3.1. (Compare Spaces, Definition 65.13.2.) Consider a big fppf site $\mathit{Sch}_{fppf} = (\mathit{Sch}/\mathop{\mathrm{Spec}}(\mathbf{Z}))_{fppf}$. Let $X$ be an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $\Delta : X \to X \times X$ be the diagonal morphism.

We say $X$ is

*separated*if $\Delta $ is a closed immersion.We say $X$ is

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

*quasi-separated*if $\Delta $ is quasi-compact.We say $X$ is

*Zariski locally quasi-separated*^{2}if there exists a Zariski covering $X = \bigcup _{i \in I} X_ i$ (see Spaces, Definition 65.12.5) such that each $X_ i$ is quasi-separated.

Let $S$ is a scheme contained in $\mathit{Sch}_{fppf}$, and let $X$ be an algebraic space over $S$. Then we say $X$ is *separated*, *locally separated*, *quasi-separated*, or *Zariski locally quasi-separated* if $X$ viewed as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ (see Spaces, Definition 65.16.2) has the corresponding property.

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