Cocountable

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.

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 on any set X consists of the empty set and all cocountable subsets of X. In the cocountable topology, the only closed subsets are countable sets, or the whole of X. Then X is automatically Lindelöf in this topology, since every open set only omits countably many points of X.