Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was doing some problems , but I don´t know how to prove 2 of them Dx , that are about homeomorphism. I have to prove that $$ R^{n + 1} - \left\{ 0 \right\} \cong S^n \times\,R $$ where R denotes the real numbers all of this with the usual topology of $R^n$

For every $c>0$ $ \left\{ {\left( {x,y,z} \right) \in R^3 :x^2 + y^2 - z^2 = c} \right\} \cong S^1 \,\times\,R $ I think that it will be useful to use the last problem :/

share|cite|improve this question

Define a map from $\mathbb R^{n+1}\setminus\{0\}$ to $S^n\times\mathbb R^+$ as follows: $$\vec{x}\mapsto (\vec{x}/||\vec{x}||,||\vec{x}||).$$ Now verify that this is a homeomorphism, and use that $\mathbb R^+\cong \mathbb R$.

share|cite|improve this answer
    
It looks like the second coordinate $||\vec{x}||$ can't be $0$, so the function isn't onto. Edit: Oops! the $+$ in $\mathbb{R}^+$ didn't show on my screen until just now. – Patrick Feb 29 '12 at 20:25
    
The idea is: the $S^n$ part tells you the "direction" of the point, and the $R$ part (which can be seen as $(0, +\infty)$) the distance from $0$, so think polar coordinates, e.g. for the plane. – Henno Brandsma Feb 29 '12 at 21:03
    
That IS the simplest way to do it. I actually recall this was a question on my general topology midterm exam a few years ago. – Mathemagician1234 Feb 29 '12 at 21:08
    
@Patrick: Sorry, that was a mistake in the first version of my answer, which I fixed. – Grumpy Parsnip Feb 29 '12 at 21:45

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.