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

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

Let $x, y \in R$.

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

Then:
 * $y \in {B_r}^- \paren x \implies {B_r}^- \paren y = {B_r}^- \paren x$

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

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

By the definition of an closed ball, then:
 * $\norm {a - y} \le r$
 * $\norm {y - x} \le r$

Hence:

By the definition of a closed ball, then:
 * $a \in {B_r}^- \paren x$.

Hence:
 * ${B_r}^- \paren y \subseteq {B_r}^- \paren x$

By Norm of Negative then:
 * $\norm {x - y} \le r$

By the definition of a closed ball, then:
 * $x \in {B_r}^- \paren y$

Similarly it follows that:
 * ${B_r}^- \paren x \subseteq {B_r}^- \paren y$

By set equality:
 * ${B_r}^- \paren x = {B_r}^- \paren y$