In mathematics, a cofinite subset whose complement is a finite set If the complement is not finite, but is countable, then one says the set is cocountable.

These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.

This use of the prefix "" to describe a property possessed by a set's complement is consistent with its use in other terms such as "comeagre set".

Boolean algebras

that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the

has a unique non-principal ultrafilter (that is, a maximal filter.

Cofinite topology

The cofinite topology or the finite complement topology is a topology It has precisely the empty set For this reason, the cofinite topology is also known as the finite-closed topology

This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable over a field (considered as affine line) is the cofinite topology. The same is true for any irreducible algebraic curve in the plane.

Properties

• Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology.

• Compactness: Since every open set is compact and sequentially compact.

• Separation: The cofinite topology is the coarsest topology satisfying the T1 axiom; that is, it is the smallest topology for which every singleton set satisfies the T1 is finite then the cofinite topology is simply the discrete topology is not finite then this topology is not Hausdorff (T2), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected).

Double-pointed cofinite topology

The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of each doublet are topologically indistinguishable. It is, however, R0 since topologically distinguishable points are separated. The space is compact as the product of two compact spaces; alternatively, it is compact because each nonempty open set contains all but finitely many points.

of integers can be given a topology such that every even number is topologically indistinguishable from the following odd number or is the empty set.

Other examples

Product topology

The product topology has basis

The analog without requiring that cofinitely many factors are the whole space is the box topology.

Direct sum

The elements of the direct sum of modules

The analog without requiring that cofinitely many summands are zero is the direct product.

See also

References

• (See example 18)