cofinite

Cofinite refers to a property of a subset of a universal set X: a subset A ⊆ X is cofinite if its complement X\A is finite. Equivalently, A is cofinite when almost all elements of X lie in A. In many contexts, cofinite is also used to describe the cofinite topology on a set X, where the open sets are precisely the empty set and those with finite complement. In a cofinite topology, the closed sets are exactly the finite subsets and X itself.

When X is infinite, the cofinite topology is strictly coarser than the discrete topology; it is T1 but not Hausdorff. Any two nonempty open sets intersect, which makes the space hyperconnected and connected; in fact every infinite subset is dense, since its closure is X. If X is finite, the cofinite topology coincides with the discrete topology, since every subset has a finite complement.

Key properties of the cofinite topology on an infinite set include compactness (every open cover has a finite subcover) and the lack of separation axioms beyond T1. The space is not regular or normal in general, reflecting the limited ability to separate disjoint closed sets by neighborhoods.

Examples help illustrate the concept: on X = N, a subset A is cofinite if N\A is finite, such as A = N \ {1,2,3}. In the language of functions, a map f: X → Y is continuous with respect to the cofinite topology on X if the preimage of every open set in Y is either empty or cofinite in X.

See also: cofinite, cofinite topology, density.