TITLE: How to show that the unit sphere is a topological manifold?
QUESTION [12 upvotes]: Sorry for this basic question, but I´m not sure of something. I want to see one example.
The definition of a n-manifold is a Hausdorff space, such that each point has an open neighborhood homeomorphic to the open n-dimensional disc.
How can i prove that $$
S^n  = \left\{ {x \in \mathbf{R}^{n + 1} :\left| x \right| = 1} \right\}
$$
 is an n-manifold

REPLY [6 votes]: Alternatively, you can use the local form of a submersion to proove tha $S^n$ is a submanifold of $\mathbb{R}^{n+1}$
http://en.wikipedia.org/wiki/Submersion_%28mathematics%29
$f:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$, $f(x_1,...,x_{n+1})=x_1^2+...+x_{n+1}^2$ is a $C^\infty$-submersion on $S^n=f^{-1}(1)$.
Another nice method is to use steregraphic projections
http://en.wikipedia.org/wiki/Stereographic_projection
$S^n=U_N\cup U_S$ where $U_N=S^n\setminus\{(0,...,0,1)\}$ and $U_S=S^n\setminus\{(0,...,0,-1)\}$ are opens for the induced topology of $\mathbb{R}^{n+1}$ on $S^n$.
One can check that : $\varphi_N : U_N\rightarrow\mathbb{R}^n$, $\varphi_N(x_1,...,x_{n+1})=\big(\dfrac{x_1}{1-x_{n+1}},...,\dfrac{x_n}{1-x_{n+1}}\big)$ and $\varphi_S : U_S\rightarrow\mathbb{R}^n$, $\varphi_S(x_1,...,x_{n+1})=\big(\dfrac{x_1}{1+x_{n+1}},...,\dfrac{x_n}{1+x_{n+1}}\big)$ are homomorphisms.