---
tag: T62369RFH6
kind: object
name: n-sphere
msc: [57N99, 51M05]
instance_of: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/N-sphere
same_as:
  wikidata: Q306610
  wikipedia: N-sphere
---

## Description

Let $n \ge 0$ be an integer.
The *$n$-sphere* is the subspace
$$S^n = \set{x \in \R^{n+1} : x_1^2 + \cdots + x_{n+1}^2 = 1}$$
of [[Euclidean space|0XPR1E85W0]] $\R^{n+1}$, the set of points at distance $1$ from the origin.
It is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.
It is [[Hausdorff|0S1FJ8VEX9]] and [[second-countable|03PPNCM9GG]] as a subspace of $\R^{n+1}$.
For the north pole $N = (0, \ldots, 0, 1)$, the stereographic projection
$$\sigma_N(x) = \frac{(x_1, \ldots, x_n)}{1 - x_{n+1}}$$
is a [[homeomorphism|0QT00PB5G2]] from $S^n \setminus \set{N}$ onto $\R^n$, with inverse $y \mapsto \bigl(2y, \abs{y}^2 - 1\bigr) / \bigl(\abs{y}^2 + 1\bigr)$.
The projection from the south pole is defined in the same way, and the two charts cover $S^n$.

## Remarks

$S^0$ is a pair of points, $S^1$ is the [[circle|060W5CBZP6]] and $S^2$ is the usual round sphere.
$S^n$ is [[compact|0TVJZYB6SH]], being closed and bounded in $\R^{n+1}$, and for $n \ge 1$ it is [[path-connected|0WB6DH499T]].
For $n \ge 2$ it is simply connected.
For $n \ge 1$ it is the one-point compactification of $\R^n$, and it is homeomorphic to the closed ball $D^n$ with its boundary sphere collapsed to a point.
Its integer homology is $\Z$ in degrees $0$ and $n$ and zero in the other degrees, for $n \ge 1$.

## Explanations

### L1 human

The $n$-sphere is the set of points at distance one from a centre, in one dimension more than the sphere itself.
The $1$-sphere is a circle in the plane, the $2$-sphere is the surface of a ball in space.
Near each point it looks like flat $n$-dimensional space, but as a whole it closes up on itself.