Definition:Closed Ball

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$.

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

The closed $\epsilon$-ball of $a$ in $M$ is defined as:
 * ${B_\epsilon}^- \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) \le \epsilon}\right\}$

where $B^-$ recalls the notation of topological closure.

If it is necessary to show the metric itself, then the notation ${B_\epsilon}^- \left({a; d}\right)$ can be used.

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

Also denoted as
The notation $B^- \left({a; \epsilon}\right)$ can be found for ${B_\epsilon}^- \left({a}\right)$, particularly when $\epsilon$ is a more complicated expression than a constant.

Similarly, some sources allow ${B_d}^- \left({a; \epsilon}\right)$ to be used for ${B_\epsilon}^- \left({a; d}\right)$.

It needs to be noticed that the two styles of notation allow a potential source of confusion, so it is important to be certain which one is meant.

Also see

 * Definition:Open Ball of Metric Space