# Definition:Sphere/Normed Division Ring

< Definition:Sphere(Redirected from Definition:Sphere in Normed Division Ring)

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## 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 **$\epsilon$-sphere of $a$ in $\struct{R, \norm{\,\cdot\,}}$** is defined as:

- $S_\epsilon \paren{a} = \set {x \in R: \norm{x - a} = \epsilon}$

### Radius

In $\map {S_\epsilon} a$, the value $\epsilon$ is referred to as the **radius** of the $\epsilon$-sphere.

### Center

In $\map {S_\epsilon} a$, the value $a$ is referred to as the **center** of the $\epsilon$-sphere.

## Also known as

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

From Sphere in Normed Division Ring is Sphere in Induced Metric, the **$\epsilon$-sphere of $a$ in $\struct {R, \norm {\,\cdot\,} }$** is the $\epsilon$-sphere of $a$ in $\struct {R, d}$.

## Also see

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 2.3$ Topology, Problem $51$