Cofinitesets

Cofinitesets, or cofinite sets, are subsets whose complements are finite. Given a universal set X, a subset A ⊆ X is cofiniteset if X\A is finite. The collection of all cofinitesets of X is often denoted Cof(X). This concept is frequently used in topology, where cofinitesets generate the cofinite topology.

In the cofinite topology on an infinite set X, the open sets are exactly the empty set and the cofinitesets. This topology is T1, since singletons are closed (the complement of {x} is X\{x}, a cofiniteset). However, it is not Hausdorff when X is infinite: any two nonempty open sets intersect because cofinitesets have finite complements, so disjoint neighborhoods are impossible. The space is compact: every open cover has a finite subcover. It is also connected (indeed hyperconnected): any two nonempty open sets intersect, and the only closed sets are finite sets and X.

Finite sets are closed in the cofiniteset topology, while infinite subsets are dense, since any nonempty open set intersects them. The intersection of finitely many cofinitesets is cofiniteset, and the union of any family of cofinitesets is also cofiniteset, making Cof(X) form a filter, known as the cofinite filter. This filter can be extended to nonprincipal ultrafilters on X.

If X is finite, every subset is cofiniteset (the topology is discrete). The cofiniteset notion thus contrasts with the familiar discrete topology and serves as a classic example of a compact, T1 space that is not Hausdorff when X is infinite.