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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 :/

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1 Answer 1

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 $\mathbb R^+\cong \mathbb R$.

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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

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