Definition:Sphere/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 $51$