Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric

Theorem

Let $p$ be a prime number.

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

Let $a \in \Q_p$.

Let $n \in \Z$.

Then the open ball $\map {B_{p^{-n}}} a$ and closed ball $\map {B^-_{p^{-n}}} a$ are clopen in the $p$-adic metric.

Proof

We begin by proving the theorem for the closed ball $\map {B^-_{p^{-n}}} a$.

From Open Ball in P-adic Numbers is Closed Ball then the theorem will be proved.

From P-adic Numbers form Non-Archimedean Valued Field::

$\norm {\,\cdot\,}_p$ is a non-Archimedean division ring norm.

By definition the $p$-adic closed ball:

$\map {B^-_{p^{-n} } } a$ is a closed ball in a normed division ring.

From Closed Ball of Non-Archimedean Division Ring is Clopen:

$\map {B^-_{p^{-n} } } a$ is clopen in the $p$-adic metric.

The result follows.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$