Quasi-separated morphism (${\Delta }_{X/S}\text{ quasi-compact}$)

A morphism of schemes $f:X\to S$ is quasi-separated if its diagonal morphism $X\to X{\times }_{S}X$ is a quasi-compact morphism. Every separated morphism has this property since a closed immersion is a quasi-compact morphism. This compactness condition is needed when uniqueness in the valuative criterion is used to deduce separatedness for arbitrary schemes.