# Definition:Open Ball

## Contents

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

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

## 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 $a$**.

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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.4$: Open Balls and Neighborhoods: Definition $4.1$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: $\epsilon$-Balls - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Definition $2.3.1$ - 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 - 1999: Theodore W. Gamelin and Robert Everist Greene:
*Introduction to Topology*(2nd ed.) ... (previous) ... (next): $\S 1.1$: Open and Closed Sets - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**open ball**