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Encyclopedia > Cofinite set

In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. In other words, Y contains all but finitely many elements of X.

## Boolean algebras

The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the finite_cofinite algebra on X. A Boolean algebra A has a unique non_principal ultrafilter (i.e. a maximal filter not generated by a singleton set) if and only if there is an infinite set X such that A is isomorphic to the finite_cofinite algebra on X. In this case, the non_principal ultrafilter is the set of all cofinite sets.

## Topology

The cofinite topology on any set X consists of the empty set and all cofinite subsets of X. In the cofinite topology, the only closed subsets are finite sets, or the whole of X. Then X is automatically compact in this topology, since every open set only omits finitely many points of X. Also, the cofinite topology is the smallest topology satisfying the T1 axiom; i.e. it is the smallest topology for which every singleton set is closed. In fact, an arbitrary topology on X satisfies the T1 axiom if and only if it contains the cofinite topology.

One place where this concept occurs naturally is in the context of the Zariski topology. Since polynomials over a field K are zero on finite sets, or the whole of K, the Zariski topology on K (considered as affine line) is the cofinite topology. The same is true for any irreducible algebraic curve; it is not true, for example, for XY = 0 in the plane. Results from FactBites:

 Compact space (802 words) That is, any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space. The Cantor set (and the p-adic integers, which are homeomorphic to the Cantor set). Any space carrying the cofinite topology[?] (a set being open iff it is empty or its complement is finite).
 T1 space - Wikipedia, the free encyclopedia (909 words) The cofinite topology on an infinite set is a simple example of a topology that is T Alternatively, the set of even integers is compact but not closed, which would be impossible in a Hausdorff space. Then the basis of the topology are given by finite intersections of the subbasis sets: given a finite set A, the open sets of X are
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