Compact Complement Topology is Separable/Proof 2

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

Then $T$ is a separable space.

Proof
We have that:
 * Rationals are Dense in Compact Complement Topology
 * Rational Numbers are Countably Infinite

Hence the result by definition of separable space.