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:


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

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