Definition:Open Ball

Definition
Let $M = \left({A, d}\right)$ 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:


 * $B_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$

If it is necessary to show the metric or pseudometric itself, then the notation $B_\epsilon \left({a; d}\right)$ 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: