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September 5th, 2016, 04:55 PM  #1 
Member Joined: Oct 2014 From: Colorado Posts: 40 Thanks: 21  Construct a homeomorphism between a circle and the real number line
So we have a circle with a hole in it and we want to show it's homeomorphic to the real number line. Let $S^1 = \{ (x,y) x^2 + (y1)^2 = 1 \}$ where the north pole is at $(0,2)$, we want to get rid of the north pole. So now we try to find a mapping $f:S^1 \setminus \{0,2\} \rightarrow \mathbb R $. One such mapping could be $f(x,y) = \frac{x}{y2}$ this way it's undefined when $y=2$. However, how do we find the continuous inverse (if it exists). The problem is now we can only plug in one number from $\mathbb R$ and have to map it back to the circle and I can't think of a way to do it.

September 5th, 2016, 06:29 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,156 Thanks: 630 
Complex exponential. Map $t \in (0, 1)$ to $(\cos 2\pi t, \ \sin 2 \pi t )$. Then use the tan/arctan to map the interval to the line. ps  Your idea works too, I didn't look at the details. Are you doing 2D stereographic projection from the north pole? Last edited by Maschke; September 5th, 2016 at 06:41 PM. 

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circle, construct, homeomorphism, homeomorphisms, line, number, real 
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