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