Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls
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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:
- $\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$
Let $x, y \in R$.
Let $r \in \R_{\gt 0}$.
Then:
- $y \in \map { {B_r}^-} x \implies \map { {B_r}^-} y = \map { {B_r}^-} x$
Proof
Let $y \in \map { {B_r}^-} x$.
Let $a \in \map { {B_r}^-} y$.
By the definition of an closed ball, then:
- $\norm {a - y} \le r$
- $\norm {y - x} \le r$
Hence:
\(\ds \norm {a - x}\) | \(=\) | \(\ds \norm {a - y + y - x}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm {a - y}, \norm {y - x} }\) | Definition of Non-Archimedean Division Ring Norm | |||||||||||
\(\ds \) | \(\le\) | \(\ds r\) |
By the definition of a closed ball, then:
- $a \in \map { {B_r}^-} x$.
Hence:
- $\map { {B_r}^-} y \subseteq \map { {B_r}^-} x$
By Norm of Negative then:
- $\norm {x - y} \le r$
By the definition of a closed ball, then:
- $x \in \map { {B_r}^-} y$
Similarly it follows that:
- $\map { {B_r}^-} x \subseteq \map { {B_r}^-} y$
By set equality:
- $\map { {B_r}^-} x = \map { {B_r}^-} y$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology, Proposition $2.3.6 \ \text {(ii)}$