Compact Complement Topology is Separable

Theorem
Let $T = \left({\R, \tau}\right)$ be the compact complement topology on $\R$.

Then $T$ is a separable space.

Proof 1

 * Compact Complement Topology is Second-Countable
 * Second-Countable Space is First-Countable and Separable

Proof 2
We have that:
 * Rationals Dense in Compact Complement Topology
 * Rational Numbers are Countable

Hence the result by definition of separable space.