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 $\norm{\,\cdot\,}$,

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

Let $x, y \in R$.

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

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

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

Then:
 * If $y \in B_r \paren x$, then $B_r \paren x = B_r \paren y$

Proof
Let $y \in B_r \paren x$.

Lemma

 * $B_r \paren y \subseteq B_r \paren x$

Proof
Now:

So $x \in B_r \paren y$ and the previous argument applies with the roles of $x$ and $y$ reversed.

Hence:
 * $B_r \paren y \subseteq B_r \paren x$

Finally, it has been shown:
 * $B_r \paren x \subseteq B_r \paren y$
 * $B_r \paren y \subseteq B_r \paren x$

So by definition of set equality:
 * $B_r \paren x = B_r \paren y$

which is what we needed to prove.