Cofinite

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(Redirected from Cofinite topology)

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.

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