Quasicompact

Quasi-compact, often written quasicompact, is a property of a topological space X defined by the open-cover condition: every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets {U_i}, then there exists a finite subcollection whose union is X. The term is particularly common in algebraic geometry, where one often works without assuming separation axioms; quasi-compactness provides a useful generalization of compactness in that setting.

In many texts, compactness and quasi-compactness are treated as the same notion, with the distinction arising mainly from historical and contextual usage. The word “quasi-compact” is favored in algebraic geometry to emphasize that separation properties (such as being Hausdorff) are not required. Nevertheless, for spaces that do satisfy the usual separation axioms, quasi-compactness and compactness coincide.

Basic properties include: closed subspaces of a quasi-compact space are quasi-compact; finite unions of quasi-compact subspaces are quasi-compact; the continuous image of a quasi-compact space is quasi-compact. Quasi-compactness is also stable under finite unions and, under the axiom of choice, under arbitrary products in the sense of Tychonoff-type results.

A central example is the spectrum of a ring, Spec A. It is quasi-compact because 1 can be generated by finitely many elements a1, ..., an, and the standard opens D(ai) form a finite cover of Spec A. In algebraic geometry, a scheme is quasi-compact if its underlying topological space is quasi-compact; equivalently, it can be covered by finitely many affine open subschemes. Quasi-compact morphisms, finiteness conditions, and related notions are developed to work effectively without assuming Hausdorffness.