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

Let $x, y \in R$.

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

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

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

If $r \le s$ then:

Similarly, if $s \le r$ then:

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

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

By the definition of an open ball then:
 * $x \in {B_r}^- \paren x \neq \empty$
 * $y \in {B_s}^- \paren y \neq \empty$

The result follows.