Cocountable topology

Definition

Suppose $X$ is a set. The cocountable topology on $X$ is a topological space structure on $X$ that can be defined in the following equivalent ways:

Type of description	Details
in terms of open subsets	The open subsets are precisely the empty set and the subsets whose complements are either finite or countable.
in terms of closed subsets	The closed subsets are precisely the whole space, the finite subsets, and the countable subsets.

If the set is finite or countable, the cocountable topology makes it a discrete space.