Sphere in Normed Division Ring is Sphere in Induced Metric

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

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

Let $a \in R$.

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

Let $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ denote the sphere in the normed division ring $\struct {R, \norm {\,\cdot\,} }$.

Let $\map {S_\epsilon} {a; d }$ denote the sphere in the metric space $\struct {R, d}$.

Then:
 * $\map {S_\epsilon} {a; \norm {\,\cdot\,} }$ = $\map {S_\epsilon} {a; d }$

Proof
The result follows from Equality of Sets.

Also see

 * Closed Ball in Normed Division Ring is Closed Ball in Induced Metric


 * Open Ball in Normed Division Ring is Open Ball in Induced Metric