Cocountability

In mathematics, a cocountable subset of a set ${\displaystyle X}$ is a subset ${\displaystyle Y}$ whose complement in ${\displaystyle X}$ is a countable set. In other words, ${\displaystyle Y}$ contains all but countably many elements of ${\displaystyle X}$. 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 ${\displaystyle Y}$ is cofinite.¹

The set of all subsets of ${\displaystyle X}$ 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 ${\displaystyle X}$. It is the smallest σ-algebra containing every singleton set.²

The cocountable topology (also called the "countable complement topology") on any set ${\displaystyle X}$ consists of the empty set and all cocountable subsets of ${\displaystyle X}$.³