# Cocountability

In mathematics, a **cocountable** subset of a set *X* is a subset *Y* whose complement in *X* is a countable set. In other words, *Y* contains all but countably many elements of *X*. While the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals. If the complement is finite, then one says *Y* is cofinite.

## σ-algebras

The set of all subsets of *X* that are either countable or cocountable forms a σ-algebra, i.e., it is closed under the operations of countable unions, countable intersections, and complementation. This σ-algebra is the **countable-cocountable algebra** on *X*. It is the smallest σ-algebra containing every singleton set.

## Topology

The cocountable topology (also called the "countable complement topology") on any set *X* consists of the empty set and all cocountable subsets of *X*.