Cofiniteness

Not to be confused with cofinality.

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 it 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.

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 single element of the algebra) 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.

Cofinite topology

The cofinite topology (sometimes called the finite complement topology) is a topology which 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 \mid A=\varnothing \mbox{ or } X \setminus A \mbox{ is finite} \}

This topology occurs naturally 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.

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; 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. If X is finite then the cofinite topology is simply the discrete topology. If X is not finite, then this topology is not T₂, regular or normal, since no two nonempty open sets are disjoint (i.e. 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. It is not T₀ or T₁, since the points of the doublet are topologically indistinguishable. It is, however, R₀ since the topologically distinguishable points are separable.

An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together. Let X be the set of integers, and let O_A be a subset of the integers whose complement is the set A. Define a subbase of open sets G_x for any integer x to be G_x = O_{{x, x+1}} if x is an even number, and G_x = O_{{x-1, x}} if x is odd. Then the basis sets of X are generated by finite intersections, that is, for finite A, the open sets of the topology are

U_A := \bigcap_{x \in A} G_x

The resulting space is not T₀ (and hence not T₁), because the points x and x + 1 (for x even) are topologically indistinguishable. The space is, however, a compact space, since it is covered by a finite union of the U_A.

Other examples

Product topology

The product topology on a product of topological spaces $\prod X_i$ has basis $\prod U_i$ where $U_i \subset X_i$ is open, and cofinitely many $U_i = X_i$ .

The analog (without requiring that cofinitely many 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 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 507446 (See example 18)

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