Let be a non empty set, and then the collection of subsets of whose compliments are finite along with (empty set) forms a topology on , and is called co-finite topology.
Let with topology is a co–finite topology because the compliments of all the subsets of are finite.
It may be noted that every infinite set may or may not be a co–finite topology. For this suppose (set of real numbers which is an infinite set) with topology is a co–finite topology because the compliments of all the members of topology along with empty set are finite.
If is finite, then the topology is discrete. For a subset of belongs to if and only if it is either empty or its compliment is finite. When is finite, the compliment of each of its subset is finite and therefore, each subset of belongs to . Hence is the discrete topology on .