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

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 $2.3.6 \, \text {(vi)}$