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Thread: 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 $\displaystyle \neq 0$, and match the point on the circle they pass through with the one where they intersect the x-axis:

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

    Tonio
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