Compact Complement Topology is Separable

Theorem

Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.

Then $T$ is a separable space.

Proof 1

We have:

Compact Complement Topology is Second-Countable

Second-Countable Space is Separable

Hence the result.

$\blacksquare$

Proof 2

We have that:

Rationals are Dense in Compact Complement Topology

Rational Numbers are Countably Infinite

Hence the result by definition of separable space.

$\blacksquare$

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: $6$