The Stacks project

Definition 67.4.2. 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.

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

2. We say $f$ is locally separated¹ if $\Delta _{X/Y}$ is an immersion.

3. We say $f$ is quasi-separated if $\Delta _{X/Y}$ is quasi-compact.