Topological Properties of Non-Archimedean Division Rings/Open 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 $B_r \paren{x}$ denote the open $r$-ball of $x$ in $\struct {R,\norm{\,\cdot\,}}$

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

By Open Ball is Open Set then $B_r \paren{x}$ is open in $d$.

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

Let $\operatorname{cl} \paren {B_r \paren{x}}$ denote the closure of $B_r \paren{x}$.

Let $y \in \operatorname{cl} \paren {B_r \paren{x}}$.

By the definition of the closure of $B_r \paren{x}$ then:
 * $\forall s \gt 0 : B_s \paren{y} \cap B_r \paren{x} \neq \empty$

In particular:
 * $B_r \paren{y} \cap B_r \paren{x} \neq \empty$

Let $z \in B_r \paren{y} \cap B_r \paren{x}$.

By Centers of Open Balls then:
 * $B_r \paren{y} = B_r \paren{z} = B_r \paren{x}$

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

Hence:
 * $\operatorname{cl} \paren {B_r \paren{x}} \subseteq B_r \paren{x}$

By Subset of Metric Space is Subset of its Closure then:
 * $B_r \paren{x} \subseteq \operatorname{cl} \paren {B_r \paren{x}}$

So by definition of set equality:
 * $\operatorname{cl} \paren {B_r \paren{x}} = B_r \paren{x}$

By Set is Closed iff Equals Topological Closure then $B_r \paren{x}$ is closed.