Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen

Theorem
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,

Let $x \in R$.

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

Let $S_r \paren{x}$ denote the $r$-sphere of $x$ in $\struct {R,\norm{\,\cdot\,}}$

Then:
 * The $r$-sphere of $x$, $S_r \paren x$, is both open and closed in the metric induced by $\norm{\,\cdot\,}$.

Proof
Now

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

By Open Balls Are Clopen then $B_r \paren x$ is both open and closed in $d$.

By Metric Induces Topology then $R \setminus B_r \paren {x}$ is is both open and closed in $d$.

By Closed Balls Are Clopen then ${B_r}^- \paren x$ is both open and closed in $d$.

By Metric Induces Topology then the intersection of a finite number of open sets is open.

Hence $S_r \paren {x}$ is open in $d$.

By Intersection of Closed Sets is Closed then $S_r \paren {x}$ is closed in $d$.

The result follows.