cofinite and cocountable topologies

The cofinite topology on a set $X$ is defined to be the topology $\mathcal{T}$ where$$ \mathcal{T} = \{A \subseteq X \mid X \setminus A \hbox{ is finite, or } A=\emptyset\}.$$ 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 topology in which the closed sets are $X$ and the countable subsets of $X$ .

The cofinite topology on $X$ is the coarsest $T_1$ topology 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.