\begin{definition}[Definition:Countable Complement Topology]
Let $S$ be an infinite set whose cardinality is usually taken to be uncountable.
Let $\tau$ be the set of subsets of $S$ defined as:
:$H \in \tau \iff \relcomp S H$ is countable, or $H = \O$
where $\relcomp S H$ denotes the complement of $H$ relative to $S$.
In this definition, countable is used in its meaning that includes finite.
Then $\tau$ is the '''countable complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is a '''countable complement space'''.
\end{definition}