cofinit

Cofinit is a term used in set theory to describe a property of subsets within a given set. A subset S of a set X is called cofinite if its complement within X is finite. The complement of S in X, denoted as X \ S or Sᶜ, consists of all elements in X that are not in S. Therefore, if X \ S is a finite set, then S is considered cofinite in X.

For example, consider the set of natural numbers N = {1, 2, 3, ...}. The subset of even numbers E = {2, 4, 6, ...} is not cofinite in N because its complement, the set of odd numbers O = {1, 3, 5, ...}, is infinite. However, if we consider the set of integers Z = {..., -2, -1, 0, 1, 2, ...}, and a subset A = {x ∈ Z | x ≥ 5}, then the complement of A in Z is the set {-4, -3, -2, -1, 0, 1, 2, 3, 4}, which is finite. Therefore, A is cofinite in Z.

The concept of cofinite sets is important in various areas of mathematics, including topology and measure theory. For instance, in topology, a topology can be defined on a set where the open sets are precisely the cofinite subsets. This is known as the cofinite topology. In measure theory, cofinite sets can play a role in defining certain types of measures or understanding the properties of sets with respect to a given measure. The collection of all cofinite subsets of a set X forms a filter on X, known as the cofinite filter.