Lemma 66.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

If $Y$ is separated and $f$ is separated, then $X$ is separated.

If $Y$ is quasi-separated and $f$ is quasi-separated, then $X$ is quasi-separated.

If $Y$ is locally separated and $f$ is locally separated, then $X$ is locally separated.

If $Y$ is separated over $S$ and $f$ is separated, then $X$ is separated over $S$.

If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then $X$ is quasi-separated over $S$.

If $Y$ is locally separated over $S$ and $f$ is locally separated, then $X$ is locally separated over $S$.

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