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

Let $x \in R$.

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

Let $B_r \paren{x}$ be the open $r$-ball of $x$ in $\struct {R,d}$

Then:
 * The open $r$-ball of $x$, $B_r \paren x$, is both open and closed.

Proof
By Open Ball is Open Set then $B_r \paren{x}$ so it remains to show that $B_r \paren{x}$ is closed.

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

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