Definition:Sphere/Metric Space/Radius

Definition

Let $M = \struct{A, d}$ be a metric space or pseudometric space.

Let $a \in A$.

Let $S_\epsilon \paren{a}$ be the $\epsilon$-sphere of $a$.

In $S_\epsilon \paren{a}$, 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.

Caution

It should be noted that the radius is not intrinsic to the $\epsilon$-sphere, so that the radius of a sphere is ambiguous.