Cofinite topology

Example

Let $X$ be a set and let $\tau$ consist of $\varnothing$ and of all sets $U \subseteq X$ whose complement $X \setminus U$ is finite. Then $\tau$ is a topology on $X$, the cofinite topology.

Remarks

The closed sets are $X$ and the finite subsets of $X$. The cofinite topology is T1: each one-point set is closed. If $X$ is infinite, any two nonempty open sets meet. Thus the space is not Hausdorff. Each space with the cofinite topology is compact.

Explanations

L1 human

In the cofinite topology, a set is open when it misses only finitely many points. On an infinite set, open sets are very large, so no two nonempty open sets are disjoint.