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:


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

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

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