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Math Help - Homeomorphism between Unit Disc and C

  1. #1
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    Homeomorphism between Unit Disc and C

    I need some help for following question,...i just saw it in a text..and wondering how can i show it....
    Question : Show that the function Ф(z) = z/(1-|z|) defines a homeomorphism between the unit disc D and C.

    Any help or hints will be very apreciated...
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    First step.

    \Phi : D \to \mathbb{C}\;,\;\Phi (z)=\dfrac{z}{1-|z|}

    is continuous. What difficulties have you found?

    Second step.

    The function is injective. What difficulties have you found?

    etc,etc.

    Please, show some work.
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  3. #3
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    i have seen the function Ф(z) is continuous. indeed it is homomorphic on D
    Ф(z) is one-to-one as Ф(z1) = Ф(z2) iff z1=z2

    but i couldnt see how will it be onto and how the Inverse will be continuous..
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    [U]
    Quote Originally Posted by kinkong View Post
    but i couldnt see how will it be onto and how the Inverse will be continuous..

    Hint : Taking modulus in

    w=\dfrac{z}{1-|z|}

    we obtain

    |z|=\dfrac{|w|}{1+|w|}

    On the other hand,

    \textrm{arg}\;w=\textrm{arg}\;z
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  5. #5
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    Thank you very much...it make sense to me....
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