Definition:Open Ball

From ProofWiki
Jump to navigation Jump to search

Definition

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.


Radius

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the radius of the open $\epsilon$-ball.


Center

In $\map {B_\epsilon} a$, the value $a$ is referred to as the center of the open $\epsilon$-ball.


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

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


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

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


Real Analysis

The definition of an open ball in the context of the real Euclidean space is a direct application of this:


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

Let $\R^n$ denote a real Euclidean space

Let $\norm \cdot$ denote the Euclidean norm.

Let $a \in \R^n$.

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


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

$\map B {a, R} = \set {x \in \R^n : \norm {x - a} < R}$


$p$-adic Numbers

The definition of an open ball in the context of the $p$-adic numbers is a direct application of the definition of an open 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 $a \in R$.

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


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

$\map {B_\epsilon} a = \set {x \in \Q_p: \norm{x - a}_p < \epsilon}$


Also known as

There are various names and notations that can be found in the literature for this concept, for example:

  • Open $\epsilon$-ball neighborhood of $a$ (and in deference to the word neighborhood the notation $\map {N_\epsilon} a$, $\map N {a, \epsilon}$ or $\map N {a; \epsilon}$ are often seen)
  • Spherical neighborhood of $a$
  • Open sphere at $a$
  • Open $\epsilon$-ball centered at $a$
  • $\epsilon$-ball at $a$.


Some sources use the \varepsilon symbol $\varepsilon$ instead of the \epsilon which is $\epsilon$.


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.


Some sources use $\epsilon B$ as a convenient shorthand for $B_\epsilon$, allowing it to be understood that $B$ is an open unit ball, but this is idiosyncratic and non-standard.


Rather than say epsilon-ball, as would be technically correct, the savvy modern mathematician will voice this as the conveniently bisyllabic e-ball, to the apoplexy of his professor. And at least one contributor to this site does not believe that nobody actually says open epsilon-ball neighborhood very often, whatever opportunities to do so may arise. Life is just too short.


The term neighborhood is usually used nowadays for a concept more general than an open ball: see Neighborhood (Metric Space).


Examples

Real Number Line Example

Consider the real number line with the usual (Euclidean) metric $\struct {\R, d}$.

Let $H \subseteq \R$ denote the closed real interval $\closedint 0 1$.

Let $d_H$ denote the metric induced on $H$ by $d$.


Let $\map {B_1} {1; d}$ denote the open ball of $\struct {\R, d}$ of radius $1$ and center is $1$.

Let $\map {B_1} {1; d_H}$ denote the open ball of $\struct {H, d_H}$ of radius $1$ and center is $1$.


Then by definition:

$\map {B_1} {1; d} = \set {x \in \R: 0 < x < 2} = \openint 0 2$

However:

$\map {B_1} {1; d_H} = \set {x \in \R: 0 < x \le 1} = \hointl 0 1$.


Also see

  • Results about open $\epsilon$-balls can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Open_Ball&oldid=576634"