Definition:Sphere/P-adic Numbers/Radius

Definition

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a \in Q_p$.

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.

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.

The adjectival form radial usually means directed along a radius.