Definition

Let $p$ be a prime number.

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

Let $a \in R$.

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

The open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is defined as:

$\map {B_\epsilon} a = \set {x \in \Q_p: \norm{x - a}_p < \epsilon}$

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the radius of the open $\epsilon$-ball.

Center

In $\map {B_\epsilon} a$, the value $a$ is referred to as the center of the open $\epsilon$-ball.

Note

By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

The definition of an open $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$ is nothing more than a specific instance of the general definition of an open ball in a normed division ring.