Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls

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 $\norm{\,\cdot\,}$.

For $a \in R$ and $\epsilon \in \R_{>0}$ let:
 * ${B_\epsilon}^- \paren{a}$ denote the closed $\epsilon$-ball of $a$ in $\struct {R,d}$

Let $x, y \in R$.

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

Then:
 * If $y \in {B_r}^- \paren x$, then ${B_r}^- \paren x = {B_r}^- \paren y$