Countable Basis for P-adic Numbers/Cosets

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:
 * $\mathcal B_p = \set{q + p^n \Z_p : q \in \Q, n \in \Z}$

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

Proof
From Closed Balls of P-adic Number:
 * $\mathcal B_p = \set{ q + p^n \Z_p : q \in \Q, n \in \Z} = \set{ \map {B^{\,-}_{p^{-n}}} q : q \in \Q, n \in \Z}$

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