Topological Properties of Non-Archimedean Division Rings/Centers of Open 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 open $\epsilon$-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 open ball, then:
 * $\norm {a - y} < r$
 * $\norm {y - x} < r$

Hence:

By the definition of an open ball:
 * $a \in \map {B_r} x$

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

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

By the definition of an open ball:
 * $x \in \map {B_r} y$

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

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