Lemma 67.8.10. Let $f : X \to Y$ be a morphism of algebraic spaces over a scheme $S$.

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

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

A fibre product of quasi-compact and quasi-separated algebraic spaces is quasi-compact and quasi-separated.

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