Cofiniteness#finite–cofinite algebra

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 {{visible anchor|finite–cofinite algebra}} on $X.$

In the other direction, 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 subsets of $X$.

The cofinite topology or 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 $X.$ For this reason, the cofinite topology is also known as the finite-closed topology. Symbolically, one writes the topology as

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

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.

• 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).

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 $\backslash Z$ 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.

The product topology on a product of topological spaces $\backslash prod\; X\_i$ has basis $\backslash prod\; U\_i$ where $U\_i\; \backslash subseteq\; X\_i$ 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.

The elements of the direct sum of modules $\backslash bigoplus\; M\_i$ are sequences $\backslash alpha\_i\; \backslash in\; M\_i$ where cofinitely many $\backslash alpha\_i\; =\; 0.$

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

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

{{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 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