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:
 * $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$

Let $x, y \in R$.

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

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

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

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

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:
 * $a \in \map { {B_r}^-} x$.

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

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

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

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

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