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

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Theorem

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


Let $x \in R$.

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

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


Then:

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


Proof

We have:

\(\ds \map {S_r} x\) \(=\) \(\ds \set {y \in R : \norm {y - x} = r}\) Definition of Sphere in Normed Division Ring
\(\ds \) \(=\) \(\ds \set {y \in R : \norm {y - x} \le r} \cap \set {y \in R : \norm{y - x} \ge r}\)
\(\ds \) \(=\) \(\ds \map { {B_r}^-} x \cap \paren {R \setminus \map {B_r} x}\) Definition of Open Ball of Normed Division Ring and Definition of Closed Ball of Normed Division Ring


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

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

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

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


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

Hence $\map {S_r} x$ is open in the metric space $\struct {R, d}$.

By Intersection of Closed Sets is Closed in Topological Space then $\map {S_r} x$ is closed in $\struct {R, d}$.

The result follows.

$\blacksquare$


Sources

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