Cocountability

In mathematics, a cocountable subset of a set

X

{\displaystyle X}

is a subset

Y

{\displaystyle Y}

whose complement in

X

{\displaystyle X}

is a countable set. In other words,

Y

{\displaystyle Y}

contains all but countably many elements of

X

{\displaystyle 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

{\displaystyle Y}

is cofinite.

σ-algebras

The set of all subsets of

X

{\displaystyle 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

{\displaystyle 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

{\displaystyle X}

consists of the empty set and all cocountable subsets of

X

{\displaystyle X}

.