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

Then:
 * The closed $r$-ball of $x$, ${B_r}^- \paren x$, 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 an closed ball in $\norm{\,\cdot\,}$ then:
 * ${B_r}^- \paren x$ is an closed ball in $d$

By Closed Ball is Closed then ${B_r}^- \paren{x}$ is closed in $d$.

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

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

By Centers of Closed Balls then:
 * ${B_r}^- \paren{y} = {B_r}^- \paren{x}$

By the definition of an open ball then:
 * $y \in {B_r} \paren{y} \subseteq {B_r}^- \paren{y} = {B_r}^- \paren{x}$

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