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.