Editing Cofiniteness (section)

==Boolean algebras==

The set of all subsets of <math>X</math> that are either finite or cofinite forms a [[Boolean algebra (structure)|Boolean algebra]], which means that it is closed under the operations of [[Union (mathematics)|union]], [[intersection]], and complementation. This Boolean algebra is the '''{{visible anchor|finite–cofinite algebra}}''' on <math>X.</math>

In the other direction, a Boolean algebra <math>A</math> 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 <math>X</math> such that <math>A</math> is isomorphic to the finite–cofinite algebra on <math>X.</math> In this case, the non-principal ultrafilter is the set of all cofinite subsets of <math>X</math>.