Rationals are Dense in Compact Complement Topology

Theorem

Let $T = \struct {\R, \tau^*}$ 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.

$\blacksquare$

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: $6$