Topological Properties of Non-Archimedean Division Rings

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

Let $x, y \in R$.

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

The following results hold: