P-adic Numbers is Second Countable Topological Space

Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

Then the topological space $\struct {\Q_p, \tau_p}$ is second-countable.

Proof

From Countable Basis for P-adic Numbers, the topological space $\struct {\Q_p, \tau_p}$ has a countable basis.

By definition, $\struct {\Q_p, \tau_p}$ is second-countable.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.7$