Definition:Closed Ball

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Definition

Metric Space

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.


Normed Division Ring

Let $\struct{R, \norm {\,\cdot\,} }$ be a normed division ring.

Let $a \in R$.

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


The closed $\epsilon$-ball of $a$ in $\struct {R, \norm {\,\cdot\,} }$ is defined as:

$\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a} \le \epsilon}$


If it is necessary to show the norm itself, then the notation $\map { {B_\epsilon}^-} {a; \norm {\,\cdot\,} }$ can be used.


Normed Vector Space

Let $\struct{X, \norm {\,\cdot\,} }$ be a normed vector space.

Let $x \in X$.

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


The closed $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$ is defined as:

$\map { {B_\epsilon}^-} x = \set {y \in X: \norm {y - x} \le \epsilon}$


Real Analysis

Let $n \ge 1$ be a natural number.

Let $\R^n$ denote real Euclidean space

Let $\left\Vert{\, \cdot \,}\right\Vert$ denote the Euclidean norm.

Let $a \in \R^n$.

Let $R > 0$ be a strictly positive real number.


The closed ball of center $a$ and radius $R$ is the subset:

$B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert \le R}\right\}$


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 a closed ball in a normed division ring:


Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a \in \Q_p$.

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


The closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$ is defined as:

$\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a}_p \le \epsilon}$


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

  • Results about closed balls can be found here.


Sources