# 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**.

(*The Elements*: Book $\text{XI}$: Definition $14$)

### Topology

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

- $\Bbb S^n = \left\{{x \in \R^{n+1} : \left|{x - y}\right| = r}\right\}$

where $|\cdot|$ 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:

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

### 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:

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

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

By the definition of the metric induced by the norm, the **$\epsilon$-sphere of $a$ in $\struct{R, \norm{\,\cdot\,}}$** is the $\epsilon$-sphere of $a$ in $\struct{R, d}$