cofinite and cocountable topologies

The cofinite topology on a set $X$is defined to be the topology $𝒯$ where

$$𝒯=\{A\subseteq X\mid X\setminus A\text{ is finite, or }A=\mathrm{\varnothing }\}.$$

In other words, the closed sets in the cofinite topology are $X$ and the finite subsets of $X$.

Analogously, the cocountable topology on $X$is defined to be the topologyin which the closed sets are $X$ and the countable subsets of $X$.

The cofinite topology on $X$ is the coarsest $T_1$ topology (http://planetmath.org/T1Space) on $X$.

The cofinite topology on a finite set $X$ is the discrete topology.Similarly, the cocountable topology on a countable set $X$ is the discrete topology.

A set $X$ together with the cofinite topology forms a compact topological space.