cofinite topology

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.

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".

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 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
$\mathcal = \.$

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 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 $G_x = O_$ if $x$ is an even number, and $G_x = O_$ 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
$U_A := \bigcap_ G_x$

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 each $U_A$ contains all but finitely many points.

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 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.

See also

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References

* {{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 reprint of 1978, isbn=978-0-486-68735-3, mr=507446, year=1995 ''(See example 18)''

Basic concepts in infinite set theory
General topology