Definition:Open Ball

From ProofWiki
Jump to navigation Jump to search

Definition

Let $M = \left({A, d}\right)$ 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:

$B_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$


If it is necessary to show the metric or pseudometric itself, then the notation $B_\epsilon \left({a; d}\right)$ can be used.


Radius

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


Center

In $B_\epsilon \left({a}\right)$, 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.


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 $\left\Vert{\cdot}\right\Vert$ 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:

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


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 $N_\epsilon \left({a}\right)$, $N \left({a, \epsilon}\right)$ or $N \left({a; \epsilon}\right)$ are often seen)
  • Spherical neighborhood of $a$
  • Open sphere at $a$
  • Open $\epsilon$-ball centered at $a$
  • $\epsilon$-ball at $a$.


The notation $B \left({a; \epsilon}\right)$ can be found for $B_\epsilon \left({a}\right)$, particularly when $\epsilon$ is a more complicated expression than a constant.

Similarly, some sources allow $B_d \left({a; \epsilon}\right)$ to be used for $B_\epsilon \left({a; d}\right)$.

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.


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).


Also see

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


Sources