# 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*. Since 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.^{[1]}

## σ-algebras[edit]

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.^{[2]}

## Topology[edit]

The cocountable topology (also called the "countable complement topology") on any set *X* consists of the empty set and all cocountable subsets of *X*.^{[3]}

## References[edit]

**^**Halmos, Paul; Givant, Steven (2009), "Chapter 5: Fields of sets",*Introduction to Boolean Algebras*, Undergraduate Texts in Mathematics, New York: Springer, pp. 24–30, doi:10.1007/978-0-387-68436-9_5, ISBN 9780387684369**^**Halmos & Givant (2009), "Chapter 29: Boolean σ-algebras", pp. 268–281, doi:10.1007/978-0-387-68436-9_29**^**James, Ioan Mackenzie (1999), "Topologies and Uniformities",*Springer Undergraduate Mathematics Series*, London: Springer: 33, doi:10.1007/978-1-4471-3994-2, ISBN 9781447139942