N-spheres as closed C^inf-manifold

[Philip2016]

Homework Statement

I need to prove that the unit n-sphere is a closed C^inf-manifold, and am not sure what to do.

Homework Equations

The unit n-sphere is defined as:
S^n = {(x_0,...,x_n) belongs to R^(n+1) | (x_0)^2+...+(x_n)^2=1}

The Attempt at a Solution

It's not a proof, but a simple example of transfering any point x of distance r to origo on the spherical coordinate system to the unit n-sphere, through the infinetly differentiable function 1/r. But this is no proof, just an example. How do I prove that a certain manifold is 1) closed/opened (has/doesn't have boundary, i.e. delta M = the zero set), 2) topological/smooth/C^k?
Thanks in advance for any help.

 

[andrewkirk]

To prove closure, I suggest you use the theorem that a set is closed iff it contains all its limit points. For given values $x_1,...,x_{n-1}$ find all limit points whose first $n-1$ coordinates are $x_1,...,x_{n-1}$, and show that all such limit points are in $S^n$.

To prove $C^\infty$, you could choose an atlas in which the hypersphere is very nearly flat within each chart, which will be the case if each chart is very small. You should be able to define an atlas in which all points within each chart lie within distance $\delta$ of one another. Then you just need to prove that, on the region of overlap between any two overlapping charts (which will be a very small region), the transition map is infinitely differentiable.

 

[micromass]

To prove closure, I suggest you use the theorem that a set is closed iff it contains all its limit points. For given values $x_1,...,x_{n-1}$ find all limit points whose first $n-1$ coordinates are $x_1,...,x_{n-1}$, and show that all such limit points are in $S^n$.

An easier method is to use that a function $f$ is continuous iff for every closed set $F$ holds that $f^{-1}(F)$ is closed.