\begin{definition}[Definition:Compact Complement Topology]
Let $T = \struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology.
Let $\tau^*$ be the set defined as:
:$\tau^* = \leftset {S \subseteq \R: S = \O \text { or } \relcomp \R S}$ is compact in $\rightset {\struct {\R, \tau} }$
where $\relcomp \R S$ denotes the complement of $S$ in $\R$.
Then $\tau^*$ is the '''compact complement topology''' on $\R$, and $T^* = \struct {\R, \tau^*}$ is the '''compact complement space''' on $\R$.
\end{definition}