# Definition:Sphere

## Definition

### Geometry

A sphere is a surface in solid geometry such that all straight lines falling upon it from one particular point inside it are equal.

In the words of Euclid:

When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.

### Topology

The $n$-dimensional sphere, or $n$-sphere, is the set:

$\Bbb S^n = \set {x \in \R^{n + 1} : \size {x - y} = r}$

where $\size {\, \cdot \, }$ denotes the Euclidean distance.

### Metric Space

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 $\epsilon$-sphere of $a$ in $M$ is defined as:

$\map {S_\epsilon} a = \set {x \in A: \map d {x, a} = \epsilon}$

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

$S_\epsilon \paren{a} = \set {x \in R: \norm{x - a} = \epsilon}$

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

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

The definition of an sphere in the context of the $p$-adic numbers is a direct application of the definition of an sphere 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 $\epsilon$-sphere of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is defined as:

$\map {S_\epsilon} a = \set {x \in \Q_p: \norm {x - a} = \epsilon}$