Cofiniteness
In mathematics, a cofinite subset of a set formula_1 is a subset formula_2 whose complement in formula_1 is a finite set. In other words, formula_2 contains all but finitely many elements of formula_5 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 formula_1 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 formula_5 A Boolean algebra formula_2 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 formula_1 such that formula_2 is isomorphic to the finite–cofinite algebra on formula_5 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 formula_5 It has precisely the empty set and all cofinite subsets of formula_1 as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of formula_5 Symbolically, one writes the topology as formula_15 This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable over a field formula_16 are zero on finite sets, or the whole of formula_17 the Zariski topology on formula_16  (considered as "affine line") is the cofinite topology. The same is true for any "irreducible" algebraic curve; it is not true, for example, for formula_19 in the plane. 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 T or T, since the points of the doublet are topologically indistinguishable. It is, however, R 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 formula_1 be the set of integers, and let formula_26 be a subset of the integers whose complement is the set formula_27 Define a subbase of open sets formula_28 for any integer formula_29 to be formula_30 if formula_29 is an even number, and formula_32 if formula_29 is odd. Then the basis sets of formula_1 are generated by finite intersections, that is, for finite formula_35 the open sets of the topology are formula_36 The resulting space is not T (and hence not T), because the points  formula_29 and formula_38 (for formula_29 even) are topologically indistinguishable. The space is, however, a compact space, since each formula_40 contains all but finitely many points. Other examples. Product topology. The product topology on a product of topological spaces formula_41 has basis formula_42 where formula_43 is open, and cofinitely many formula_44 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 formula_45 are sequences formula_46 where cofinitely many formula_47 The analog (without requiring that cofinitely many are zero) is the direct product.