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Math Help - Homeomorphism

  1. #1
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    Homeomorphism

    Show that 'Unit circle\{(0,1)} is homeomorphic to R'...

    I need the function..Thank you in advance..
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  2. #2
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    Quote Originally Posted by math.dj View Post
    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
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