Topological Properties of Non-Archimedean Division Rings/Intersection 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 $r, s \in \R_{>0}$.

Then:
 * $\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$

Necessary Condition
Let $z \in \map { {B_r}^-} x \cap \map { {B_s}^-} y$.

If $r \le s$ then:

Similarly, if $s \le r$ then:

Sufficient Condition
Let:
 * $\map { {B_r}^-} x \subseteq \map { {B_s}^-} y$

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

By the definition of an open ball then:
 * $x \in \map { {B_r}^-} x \ne \O$
 * $y \in \map { {B_s}^-} y \ne \O$

The result follows.