1. ## Topology: Connectedness

Hello!

Can anybody help me with the following exercise?

Prove that the one-point compactification of $\displaystyle \mathbb{R}^{n}$ is homeomorphic to the n-sphere
$\displaystyle S^n$ = {($\displaystyle y_1$, ... , $\displaystyle y_{n+1}$ ) $\displaystyle \in \mathbb{R}^{n+1}$ : $\displaystyle y^2_1 + ... + y^2_{n+1}$ = 1}.

Hint: It suffices to show that $\displaystyle S^n$\(0,0,...,0,1) is homeomorphic to $\displaystyle \mathbb{R}^{n}$.

From a different exercise I know that the circle in $\displaystyle R^2$ is homeomorphic with R. Can I use it somehow?

2. Originally Posted by sssitex
Hello!

Can anybody help me with the following exercise?

Prove that the one-point compactification of $\displaystyle \mathbb{R}^{n}$ is homeomorphic to the n-sphere
$\displaystyle S^n$ = {($\displaystyle y_1$, ... , $\displaystyle y_{n+1}$ ) $\displaystyle \in \mathbb{R}^{n+1}$ : $\displaystyle y^2_1 + ... + y^2_{n+1}$ = 1}.

Hint: It suffices to show that $\displaystyle S^n$\(0,0,...,0,1) is homeomorphic to $\displaystyle \mathbb{R}^{n}$.

From a different exercise I know that the circle in $\displaystyle R^2$ is homeomorphic with R. Can I use it somehow?

Personally, I don't like that hint. Note though that you need only prove that $\displaystyle f:\mathbb{S}^n\to\left(\mathbb{R}^n\right)_{\infty }$ is bijective and continuous since $\displaystyle \mathbb{S}^n$ is compact and $\displaystyle \left(\mathbb{R}_n\right)_{\infty}$ is Hausdorff (since $\displaystyle \mathbb{R}^n$ is locally compact).

P.S. Define the mapping in the obvious way

3. Originally Posted by Drexel28
Personally, I don't like that hint. Note though that you need only prove that $\displaystyle f:\mathbb{S}^n\to\left(\mathbb{R}^n\right)_{\infty }$ is bijective and continuous since $\displaystyle \mathbb{S}^n$ is compact and $\displaystyle \left(\mathbb{R}_n\right)_{\infty}$ is Hausdorff (since $\displaystyle \mathbb{R}^n$ is locally compact).

P.S. Define the mapping in the obvious way

But how about the mapping (calculus) of a n-Sphere to $\displaystyle \mathbb{R}^n$? I know it is defined by the intersections of the line that goes through (0,0,....1) and the plane. But what is the parametrization?

4. Originally Posted by CSM
But how about the mapping (calculus) of a n-Sphere to $\displaystyle \mathbb{R}^n$? I know it is defined by the intersections of the line that goes through (0,0,....1) and the plane. But what is the parametrization?