# Homeomorphism

• October 31st 2009, 08:52 PM
math.dj
Homeomorphism
Show that 'Unit circle\{(0,1)} is homeomorphic to R'...

I need the function..Thank you in advance..
• October 31st 2009, 11:20 PM
tonio
Quote:

Originally Posted by math.dj
Show that 'Unit circle\{(0,1)} is homeomorphic to R'...

I need the function..Thank you in advance..

Stereographic projection: take al the "rays" (in fact, straight lines) through (0,1) with slope $\neq 0$, and match the point on the circle they pass through with the one where they intersect the x-axis:

if the line is $y=mx+1\,,\,\,m\neq 0$, then the intersection with the circle is at the point $\left(-\frac{2m}{1+m^2}\,,\,\,\frac{1-m^2}{1+m^2}\right)$ , and the intersection with the x-axis is $\left(-\frac{1}{m}\,,\,\,0\right)$.

Tonio