Homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$

Asked Dec 09 '14 at 11:10

Active Dec 09 '14 at 11:10

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I need to show that there exists a homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$

If we map $(\cos x, \sin x,y) \to (y\cos x,y\sin x)$ then only problem is that it is taking value $(0,0)$. Can we modify this or will any other map work?

asked Dec 09 '14 at 11:10

Mathronaut

5,200

Find a homeomorphism $\mathbb R \to \mathbb R^+$. – Dec 09 '14 at 11:12
2

okay $e^x$ will work,then? – Mathronaut Dec 09 '14 at 11:15
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Is This map works :$(\cos x, \sin x,y) \to (e^y\cos x,e^y\sin x)$ – erfan soheil Dec 09 '14 at 11:26

Homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$

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