[[Topology MOC]]
# Sphere space

The **$n$-sphere** $\mathbb{S}^n$ is usually defined as the boundary of the unit ball in $(n+1)$-dimensional [[Real coördinate space]] $\mathbb{R}^{n+1}$, #m/def/topology 
i.e.
$$
\begin{align*}
\mathbb{S}^n = \{ v \in \mathbb{R}^{n+1} : \|n\| = 1 \} \sube \mathbb{R}^{n+1}
\end{align*}
$$
along with the subspace metric and [[subspace topology]].
It is thereby an $n$-dimensional [[Topological manifold]],
and forms the boundary of the $(n+1)$-[[Ball space]].

## Properties

- The $n$-sphere is a [[Compact space]].
- [[Brouwer's fixed point theorem|Retraction theorem]]: No retraction from ball to boundary sphere
- [[Borsuk-Ulam theorem]]: $f : \mathbb{S}^n \to \mathbb{R}^n$ has a set of equal-valued antipodes 

### 1-sphere

- [[Degree of a circle endomorphism]]

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#state/tidy | #lang/en | #SemBr