Definition:Closed Ball

Definition
Let $M = \struct {A, d}$ 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:
 * $\map { {B_\epsilon}^-} a := \set {x \in A: \map d {x, a} \le \epsilon}$

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

If it is necessary to show the metric itself, then the notation $\map { {B_\epsilon}^-} {a; d}$ 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 $\map {B^-} {a; \epsilon}$ can be found for $\map { {B_\epsilon}^-} a$, particularly when $\epsilon$ is a more complicated expression than a constant.

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

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