# Homeomorphism between Unit Disc and C

• May 1st 2011, 10:15 PM
kinkong
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...
• May 1st 2011, 11:31 PM
FernandoRevilla
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.

• May 2nd 2011, 03:44 PM
kinkong
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..
• May 2nd 2011, 11:34 PM
FernandoRevilla
[U]
Quote:

Originally Posted by kinkong
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,

$\textrm{arg}\;w=\textrm{arg}\;z$
• May 3rd 2011, 12:56 AM
kinkong
Thank you very much...it make sense to me....