# Thread: Homeomorphism between Unit Disc and C

1. ## 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...

2. First step.

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

is continuous. What difficulties have you found?

Second step.

The function is injective. What difficulties have you found?

etc,etc.

3. 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..

4. [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

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

we obtain

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

On the other hand,

$\displaystyle \textrm{arg}\;w=\textrm{arg}\;z$

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