Topological Properties of Non-Archimedean Division Rings/Intersection 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 $r, s \in \R_{>0}$.
Then:
- $\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$
Proof
Necessary Condition
Let $z \in \map { {B_r}^-} x \cap \map { {B_s}^-} y$.
If $r \le s$ then:
\(\ds \map { {B_r}^-} x\) | \(=\) | \(\ds \map { {B_r}^-} z\) | Every element in an open ball is the center | |||||||||||
\(\ds \) | \(\subseteq\) | \(\ds \map { {B_s}^-} z\) | as $r \le s$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map { {B_s}^-} y\) | Every element in an open ball is the center |
Similarly, if $s \le r$ then:
\(\ds \map { {B_s}^-} y\) | \(\subseteq\) | \(\ds \map { {B_r}^-} x\) |
$\Box$
Sufficient Condition
Let:
- $\map { {B_r}^-} x \subseteq \map { {B_s}^-} y$
or:
- $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$
By the definition of an open ball then:
- $x \in \map { {B_r}^-} x \ne \O$
- $y \in \map { {B_s}^-} y \ne \O$
The result follows.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology, Proposition $2.3.6 \, \text {(vi)}$