nLab cofinite topology

Definition

Given a set $X$, then the cofinite topology or finite complement topology on $X$ is the topology whose open subsets are precisely

1. all cofinite subsets;

2. the empty set.

Properties

If $X$ is a finite set, then its cofinite topology coincides with its discrete topology.

The cofinite topology on a set $X$ is the coarsest topology on $X$ that satisfies the $T_1$ separation axiom, hence the condition that every singleton subset is a closed subspace.

Indeed, every $T_1$-topology on $X$ has to be finer that the cofinite topology.

If $X$ is not finite, then its cofinite topology is not sober, hence in particular not Hausdorff (since Hausdorff implies sober).

A set equipped with the cofinite topology forms a compact space. However, this type of compact space is not uniformizable; if it were, then under the $T_1$ condition it would also be Hausdorff, which as we saw is not the case.

References

• Wikipedia, Cofiniteness – Cofinite topology