# Definition:Ball

## Definition

### Open Ball

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.

### Closed Ball

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.

### Unit Ball

Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$.

The closed unit ball of $V$, denoted $\operatorname {ball} V$, is the set:

$\set {v \in V: \norm v_V \mathop \le 1}$

## Also known as

Some sources use the term disk (or disc in British English), but $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve this term for a disk in the complex plane, as there is an intuitive $2$-dimensional nuance to the word disk.