Cofiniteness

In mathematics, a cofinite subset of a set $X$ is a subset $A$ whose complement in $X$ is a finite set. In other words, $A$ contains all but finitely many elements of $X.$ 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 "co" to describe a property possessed by a set's complement is consistent with its use in other terms such as "comeagre set".

Boolean algebras

The set of all subsets of $X$ 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 finite–cofinite algebra on $X.$ A Boolean algebra $A$ 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 $X$ 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.

Cofinite topology

The cofinite topology (sometimes called the finite complement topology) is a topology that can be defined on every set $X.$ It has precisely the empty set and all cofinite subsets of $X$ as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of $X.$ Symbolically, one writes the topology as

{\mathcal {T}}=\{A\subseteq X:A=\varnothing {\mbox{ or }}X\setminus A{\mbox{ is finite}}\}.

This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable 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.

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 $X,$ the space $X$ 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. In fact, an arbitrary topology on $X$ satisfies the T₁ axiom if and only if it contains the cofinite topology. If $X$ is finite then the cofinite topology is simply the discrete topology. If $X$ is not finite then this topology is not Hausdorff (T₂), regular or normal because no two nonempty open sets are disjoint (that is, it is 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 or T₁, since the points of each doublet are topologically indistinguishable. It is, however, R₀ since topologically distinguishable points are separated. The space is 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 $\mathbb {Z}$ of integers can be given a topology such that every even number $2n$ is topologically indistinguishable from the following odd number $2n+1$ . The closed sets are the unions of finitely many pairs $2n,2n+1,$ 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 $2n,2n+1,$ or is the empty set.

Other examples

Product topology

The product topology on a product of topological spaces $\prod X_{i}$ has basis $\prod U_{i}$ where $U_{i}\subseteq X_{i}$ is open, and cofinitely many $U_{i}=X_{i}.$

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 $\bigoplus M_{i}$ are sequences $\alpha _{i}\in M_{i}$ where cofinitely many $\alpha _{i}=0.$

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

References

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 (See example 18)

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