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 $r \in \R_{>0}$.
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
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology, Problem $51$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 2.1$ Elementary topological properties, Proposition $2.6$