Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls

From ProofWiki
Jump to navigation Jump to search

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 open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$


Let $x, y \in R$.

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

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

Retrieved from "https://proofwiki.org/w/index.php?title=Topological_Properties_of_Non-Archimedean_Division_Rings/Intersection_of_Open_Balls&oldid=483763"