# Definition:Sphere/Topology

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## Definition

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.

### Radius

The **radius** of an $n$-sphere:

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

is the value $r$.

### Center

The **center** of an $n$-sphere:

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

is the point $y \in \R^{n+1}$.

### Unit Sphere

The $n$-dimensional **unit sphere**, or **unit $n$-sphere**, is the $n$-sphere of radius $1$ and center the origin:

- $\Bbb S^n = \left\{{x \in \R^{n+1} : \left\lvert{x}\right\rvert = 1}\right\}$

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**sphere**