Definition:Open Ball

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

Let $a \in A$.

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

The open $\epsilon$-ball of $a$ in $M$ is defined as:


 * $\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$

If it is necessary to show the metric or pseudometric itself, then the notation $\map {B_\epsilon} {a; d}$ can be used.

Real Analysis
The definition of an open ball in the context of the real Euclidean space is a direct application of this:

$p$-adic Numbers
The definition of an open ball in the context of the $p$-adic numbers is a direct application of the definition of an open ball in a normed division ring: