\begin{definition}[Definition:Countable Complement Topology/Countable]
It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space.
This is why the countable complement topology is usually understood to apply to uncountable sets only.
Category:Definitions/Countable Complement Topology
\end{definition}