Definition:Closed Ball
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 Euclidean Space
Let $n \ge 1$ be a natural number.
Let $\R^n$ denote the real Euclidean space of dimension $n$.
Let $\norm {\, \cdot \,}$ denote the Euclidean norm.
Let $a \in \R^n$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
The closed (Euclidean) ball of $\R^n$ of center $a$ and radius $\epsilon$ is the subset:
- $\map { { {\overline B}_\epsilon}^n} a = \set {x \in \R^n : \norm {x - a} \le \epsilon}$
If the dimension $n$ has already been established, then it is commonplace to simplify the notation and present it as $\map { {\overline B}_\epsilon} a$.
P-adic Numbers
The definition of a 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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces