# Definition:Open Ball/Normed Division Ring

## Contents

## Definition

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.

Let $a \in R$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

The **open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$** is defined as:

- $\map {B_\epsilon} a = \set {x \in R: \norm{x - a} < \epsilon}$

Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.

By the definition of the metric induced by the norm, the **open $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$** is the open $\epsilon$-ball of $a$ in $\struct {R, d}$.

If it is necessary to show the norm itself, then the notation $\map {B_\epsilon} {a; \norm {\,\cdot\,} }$ can be used.

### Radius

In $B_\epsilon \paren{a}$, the value $\epsilon$ is referred to as the **radius** of the open $\epsilon$-ball.

### Center

In $B_\epsilon \paren{a}$, the value $a$ is referred to as the **center** of the open $\epsilon$-ball.

## Also see

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 2.3$ Topology, Proposition $2.3.5$