Separated morphism of schemes

Let $f:X\to S$ be a morphism of schemes and let ${\mathrm{\Delta }}_{X\mathrm{/}S}:X\to X{\times }_{S}X$ be its diagonal morphism (The diagonal morphism).

• $f$ is separated if ${\mathrm{\Delta }}_{X\mathrm{/}S}$ is a closed immersion (Closed immersions of schemes).
• $f$ is quasi-separated if ${\mathrm{\Delta }}_{X\mathrm{/}S}$ is a quasi-compact morphism (Quasi-compact and quasi-separated morphisms).

A scheme $Y$ is called separated, respectively quasi-separated, when the structure morphism $Y\to \mathrm{Spec}\mathbb{Z}$ is separated, respectively quasi-separated. The affine-intersection condition recorded under Quasi-compact and quasi-separated morphisms is the earlier definition; its equivalence to quasi-compactness of ${\mathrm{\Delta }}_{X\mathrm{/}S}$ is proved in Quasi-separatedness and the diagonal on this page. Both properties are properties of the morphism $f$ and of its diagonal, and neither is a condition on the point-set topology of $X$ alone.