Definition:Compact Complement Topology

Definition

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 $\R$.

Also see

• Compact Complement Topology is Topology

• Results about the compact complement topology can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $22$. Compact Complement Topology