Editing Cofiniteness

{{short description|Being a subset whose complement is a finite set}}
{{Distinguish|cofinality}}
In [[mathematics]], a '''cofinite''' [[subset]] of a set <math>X</math> is a subset <math>A</math> whose [[Complement (set theory)|complement]] in <math>X</math> is a [[finite set]]. In other words, <math>A</math> contains all but finitely many elements of <math>X.</math> 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|product topology]] or [[#Direct sum|direct sum]]. 

This use of the prefix "'''{{em|co}}'''" to describe a property possessed by a set's [[Complement (set theory)|'''{{em|co}}'''mplement]] is consistent with its use in other terms such as "[[Comeagre set|'''{{em|co}}'''meagre set]]". 

==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>.

==Cofinite topology==

The '''cofinite topology''' or the '''finite complement topology''' is a [[Topological space|topology]] that can be defined on every set <math>X.</math> It has precisely the [[empty set]] and all cofinite subsets of <math>X</math> as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of <math>X.</math> For this reason, the cofinite topology is also known as the '''finite-closed topology'''. Symbolically, one writes the topology as
<math display=block>\mathcal{T} = \{A \subseteq X : A = \varnothing \mbox{ or } X \setminus A \mbox{ is finite} \}.</math>

This topology occurs naturally in the context of the [[Zariski topology]]. Since [[polynomial]]s in one variable over a [[Field (mathematics)|field]] <math>K</math> are zero on finite sets, or the whole of <math>K,</math> the Zariski topology on <math>K</math> (considered as ''affine line'') is the cofinite topology. The same is true for any ''[[Irreducible component|irreducible]]'' [[algebraic curve]]; it is not true, for example, for <math>XY = 0</math> in the plane.

===Properties===

* 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 <math>X,</math> the space <math>X</math> is [[Compact set|compact]] and [[sequentially compact]].
* Separation:  The cofinite topology is the [[Comparison of topologies|coarsest topology]] satisfying the [[T1 space|T<sub>1</sub> axiom]]; that is, it is the smallest topology for which every [[singleton set]] is closed. In fact, an arbitrary topology on <math>X</math> satisfies the T<sub>1</sub> axiom if and only if it contains the cofinite topology. If <math>X</math> is finite then the cofinite topology is simply the [[Discrete space|discrete topology]]. If <math>X</math> is not finite then this topology is not [[Hausdorff space|Hausdorff (T<sub>2</sub>)]], [[Regular space|regular]] or [[Normal space|normal]] because no two nonempty open sets are disjoint (that is, it is [[Hyperconnected space|hyperconnected]]).

===Double-pointed cofinite topology===

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 [[T0 space|T<sub>0</sub>]] or [[T1 space|T<sub>1</sub>]], since the points of each doublet are [[topologically indistinguishable]].  It is, however, [[R0 space|R<sub>0</sub>]] since topologically distinguishable points are [[Separated sets|separated]].  The space is [[compact (topology)|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 <math>\Z</math> of integers can be given a topology such that every [[even number]] <math>2n</math> is [[topologically indistinguishable]] from the following [[odd number]] <math>2n+1</math>.  The closed sets are the unions of finitely many pairs <math>2n,2n+1,</math> 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 <math>2n,2n+1,</math> or is the empty set.

==Other examples==

===Product topology===
The [[product topology]] on a product of topological spaces <math>\prod X_i</math> has [[Basis (topology)|basis]] <math>\prod U_i</math> where <math>U_i \subseteq X_i</math> is open, and cofinitely many <math>U_i = X_i.</math>

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

===Direct sum===
The elements of the [[direct sum of modules]] <math>\bigoplus M_i</math> are sequences <math>\alpha_i \in M_i</math> where cofinitely many <math>\alpha_i = 0.</math>

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

==See also==

* {{annotated link|Fréchet filter}}
* {{annotated link|List of topologies}}

==References==

{{reflist}}

* {{Citation|last1=Steen|first1=Lynn Arthur|author1-link=Lynn Arthur Steen|last2=Seebach|first2=J. Arthur Jr.|author2-link=J. Arthur Seebach, Jr.|title=[[Counterexamples in Topology]]|orig-year=1978|publisher=[[Springer-Verlag]]|location=Berlin, New York|edition=[[Dover Publications|Dover]] reprint of 1978|isbn=978-0-486-68735-3|mr=507446|year=1995}} ''(See example 18)''

[[Category:Basic concepts in infinite set theory]]
[[Category:General topology]]