Definition:Sphere/Normed Division Ring/Radius
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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.
Let $\map {S_\epsilon} a$ be the $\epsilon$-sphere of $a$.
In $\map {S_\epsilon} a$, the value $\epsilon$ is referred to as the radius of the $\epsilon$-sphere.
Linguistic Note
The plural of radius is radii, pronounced ray-dee-eye.
This irregular plural form stems from the Latin origin of the word radius, meaning ray.
The ugly incorrect form radiuses can apparently be found, but rarely in a mathematical context.
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.3$ Topology, Problem $51$