Thread: 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...

First step.



is continuous. What difficulties have you found?

Second step.

The function is injective. What difficulties have you found?

etc,etc.

Please, show some work.

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..

[U]

Originally Posted by kinkong

but i couldnt see how will it be onto and how the Inverse will be continuous..

Hint : Taking modulus in



we obtain



On the other hand,

Thank you very much...it make sense to me....