Rationals are Dense in Compact Complement Topology

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

Let $\Q$ be the set of rational numbers.

Then $\Q$ is everywhere dense in $T$.

Proof
We have that the Compact Complement Topology is Coarser than Euclidean Topology.

The result follows from Denseness Preserved in Coarser Topology.