Rational Number Space is Second-Countable

Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.

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

Proof
From Rational Numbers form Metric Space, $\struct {\Q, \tau_d}$ is a metric space.

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

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