# Cofiniteness

In mathematics, a **cofinite** subset of a set is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of 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 "** co**" to describe a property possessed by a set's

**mplement is consistent with its use in other terms such as "**

*co***meagre set".**

*co*## Boolean algebras

[edit]The set of all subsets of 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 **finite–cofinite algebra** on

In the other direction, a Boolean algebra has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite set such that is isomorphic to the finite–cofinite algebra on In this case, the non-principal ultrafilter is the set of all cofinite subsets of .

## Cofinite topology

[edit]The **cofinite topology** (sometimes called the **finite complement topology**) is a topology that can be defined on every set It has precisely the empty set and all cofinite subsets of as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of Symbolically, one writes the topology as

This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable over a field are zero on finite sets, or the whole of the Zariski topology on (considered as *affine line*) is the cofinite topology. The same is true for any *irreducible* algebraic curve; it is not true, for example, for in the plane.

### Properties

[edit]- Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology.
- Compactness: Since every open set contains all but finitely many points of the space is compact and sequentially compact.
- Separation: The cofinite topology is the coarsest topology satisfying the T
_{1}axiom; that is, it is the smallest topology for which every singleton set is closed. In fact, an arbitrary topology on satisfies the T_{1}axiom if and only if it contains the cofinite topology. If is finite then the cofinite topology is simply the discrete topology. If is not finite then this topology is not Hausdorff (T_{2}), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected).

### Double-pointed cofinite topology

[edit]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 T_{0} or T_{1}, since the points of each doublet are topologically indistinguishable. It is, however, R_{0} 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.

For an example of the countable double-pointed cofinite topology, the set of integers can be given a topology such that every even number is topologically indistinguishable from the following odd number . The closed sets are the unions of finitely many pairs or the whole set. The open sets are the complements of the closed sets; namely, each open set consists of all but a finite number of pairs or is the empty set.

## Other examples

[edit]### Product topology

[edit]The product topology on a product of topological spaces has basis where is open, and cofinitely many

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

### Direct sum

[edit]The elements of the direct sum of modules are sequences where cofinitely many

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

## See also

[edit]- Fréchet filter – frechet filter
- List of topologies – List of concrete topologies and topological spaces

## References

[edit]- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446*(See example 18)*