Topological Properties of Non-Archimedean Division Rings

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 $\struct {R, \norm{\,\cdot\,} }$

$\map { {B_\epsilon}^-} a$ denote the closed $\epsilon$-ball of $a$ in $\struct {R, \norm{\,\cdot\,}}$

Let $x, y \in R$.

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

The following results hold:

Centers of Open Balls

$y \in \map {B_r} x \implies \map {B_r} y = \map {B_r} x$

Centers of Closed Balls

$y \in \map { {B_r}^-} x \implies \map { {B_r}^-} y = \map { {B_r}^-} x$

Open Balls are Clopen

The open $r$-ball of $x$, $\map {B_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.

Closed Balls are Clopen

The closed $r$-ball of $x$, $\map { {B_r}^-} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.

Spheres are Clopen

The $r$-sphere of $x$, $\map {S_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.

Intersection of Open Balls

$\map {B_r} x \cap \map {B_s} y \ne \O \iff \map {B_r} x \subseteq \map {B_s} y$ or $\map {B_s} y \subseteq \map {B_r} x$

Intersection of Closed Balls

$\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology: Proposition $2.3.6$