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 $\norm {\,\cdot\,}_p$. For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon^-} a$ denote the closed $\epsilon$-ball of $a$.

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

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

Proof
For any $\epsilon \in \R_{>0}$ and $a \in \Q_p$ let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$. 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 Countable Basis for P-adic Numbers:
 * $\mathcal B_p$ is a countable basis for $\struct{\Q_p, \tau_p}$.