The Stacks project

Lemma 67.4.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

1. If $X$ is separated then $X$ is separated over $S$.

2. If $X$ is locally separated then $X$ is locally separated over $S$.

3. If $X$ is quasi-separated then $X$ is quasi-separated over $S$.

Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. If $X$ is separated over $S$ then $f$ is separated.

2. If $X$ is locally separated over $S$ then $f$ is locally separated.

3. If $X$ is quasi-separated over $S$ then $f$ is quasi-separated.