Definition:Sphere/Normed Division Ring

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

• Definition:Open Ball of Normed Division Ring

• Definition:Closed Ball of Normed Division Ring

• Sphere in Normed Division Ring is Sphere in Induced Metric

Sources

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