Topological Properties of Non-Archimedean Division Rings/Closed Balls are Clopen

Theorem

Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$,

Let $x \in R$.

Let $r \in \R_{\gt 0}$.

Let $\map { {B_r}^-} x$ be the closed $r$-ball of $x$ in $\struct {R,\norm {\,\cdot\,} }$

Then:

The closed $r$-ball of $x$, $\map { {B_r}^-} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.

Proof

Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.

By the definition of a closed ball in $\norm {\,\cdot\,}$ then:

$\map { {B_r}^-} x$ is a closed ball in the metric space $\struct {R, d}$.

By Closed Ball is Closed in Metric Space then $\map { {B_r}^-} c$ is closed in $d$.

So it remains to show that $\map { {B_r}^-} x$ is open in $d$.

Let $y \in \map { {B_r}^-} x$.

By Centers of Closed Balls then:

$\map { {B_r}^-} y = \map { {B_r}^-} x$

By the definition of an open ball then:

$y \in \map {B_r} y \subseteq \map { {B_r}^-} y = \map { {B_r}^-} x$

By the definition of an open set in a metric space, $\map { {B_r}^-} x$ is open.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology, Proposition $2.3.6 \, \text {(iv)}$