Definition:Compact Complement Topology

Definition
Let $T = \left({\R, \tau}\right)$ be the real number line under the usual (Euclidean) topology.

Let $\tau^*$ be the set defined as:
 * $\tau^* = \left\{{S \subseteq \R: S = \varnothing \text { or } \complement_\R \left({S}\right)}\right.$ is compact in $\left.{\left({\R, \tau}\right)}\right\}$

where $\complement_\R \left({S}\right)$ denotes the complement of $S$ in $\R$.

Then $\tau^*$ is the compact complement topology on $\R$, and $T^* = \left({\R, \tau^*}\right)$ is the compact complement space on $\R$.

Also see

 * Compact Complement Topology is Topology