Countable Basis for P-adic Numbers/Closed Balls

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:
 * $\mathcal B_p = \set{ \map q : q \in \Q, n \in \Z}$

is a countable basis for $\struct{\Q_p, \tau_p}$.

Proof
From Open Ball in P-adic Numbers is Closed Ball:
 * $\mathcal B_p = \set{ \map {B^{\,-}_{p^{-n}}} q : q \in \Q, n \in \Z} = \set{ \map {B_{p^{-n+1}}} q : q \in \Q, n \in \Z} = \set{ \map {B_{p^{-n}}} q : q \in \Q, n \in \Z}$

From Leigh.Samphier/Sandbox/Countable Basis for P-adic Numbers:
 * $\mathcal B_p$ is a countable basis for $\struct{\Q_p, \tau_p}$.