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September 5th, 2016, 03:55 PM   #1
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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 $S^1 = \{ (x,y)| x^2 + (y-1)^2 = 1 \}$ 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 $f(x,y) = \frac{x}{y-2}$ 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.
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September 5th, 2016, 05:29 PM   #2
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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 05:41 PM.
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