In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X×YX is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi compact). A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated. Quasi-separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by Grothendieck (1964, 1.2.1) as a generalization of separated morphisms.
All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated. Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated.
The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi-compact.