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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $30$. The Rational Numbers: $7$