Definition:Ball

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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.


This needs considerable tedious hard slog to complete it.
In particular: Clapham and Nicholson distinguish between a disc, which is what it is in its context of a circle in a plane, and a disk, which is used as a synonym for an open or closed ball in a metric space.
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Also see


Sources

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