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 $\R^{n+1}$, the set of points at distance $1$ from the origin. It is a topological manifold of dimension $n$. It is Hausdorff and second-countable 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 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 $S^n$.