\begin{definition}[Definition:Finite Complement Topology]
Let $S$ be a set whose cardinality is usually specified as being infinite.
Let $\tau$ be the set of subsets of $S$ defined as:
:$H \in \tau \iff \relcomp S H \text { is finite, or } H = \O$
where $\relcomp S H$ denotes the complement of $H$ relative to $S$.
Then $\tau$ is the '''finite complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is a '''finite complement space'''.
\end{definition}