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Concept
Cofiniteness
Summary
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 "" to describe a property possessed by a set's mplement is consistent with its use in other terms such as "meagre set".
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 on 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 sets.
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.
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 T1 axiom; that is, it is the smallest topology for which every singleton set is closed.
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