Editing Cofiniteness (section)

==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.