
Topology Topology Math Forum 
 LinkBack  Thread Tools  Display Modes 
September 5th, 2016, 03:55 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, 05:29 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,254 Thanks: 293 
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. 

Tags 
circle, construct, homeomorphism, homeomorphisms, line, number, real 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
irrational number multiplied by any real number  mick7  Number Theory  13  July 13th, 2015 10:08 PM 
Is the completeness of the real number line equivalent to Dedekind's axiom?  RyanPowers  Real Analysis  7  September 25th, 2014 10:55 AM 
Fractions and the Real Line  mdocka1  Elementary Math  2  January 10th, 2014 05:48 PM 
Real line bundles over s1  everk  Real Analysis  0  January 29th, 2012 11:54 AM 
Finding real number in complex number  TsAmE  Complex Analysis  1  October 18th, 2010 04:38 PM 