Irrational Number Space is Second-Countable

Theorem

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.

Then $\struct {\R \setminus \Q, \tau_d}$ is second-countable.

Proof

From Irrational Numbers form Metric Space, $\struct {\R \setminus \Q, \tau_d}$ is a metric space.

From Irrational Number Space is Separable, $\struct {\R \setminus \Q, \tau_d}$ is a separable space.

The result follows from Separable Metric Space is Second-Countable.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $7$