Topological Properties of Non-Archimedean Division Rings/Intersection of Open 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 $\norm{\,\cdot\,}$.

Let $x, y \in R$.

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

Let $B_r \paren{x}$ be the open $r$-ball of $x$ in $\struct {R,d}$

Let $B_s \paren{y}$ be the open $s$-ball of $y$ in $\struct {R,d}$

Then:
 * ${B_r} \paren x \cap {B_s} \paren y \ne \empty$ ${B_r} \paren x \subseteq {B_s} \paren y$ or ${B_r} \paren x \supseteq {B_s} \paren y$