# Thread: Complex Analysis - simply connected domain

1. ## Complex Analysis - simply connected domain

Prove that if domain $\Omega$ conformaly maps to unit disc $D=\left \{ \left | z \right |<1 \right \}$then $\Omega$ it's simply connected domain.

Thanks!

2. ## Re: Complex Analysis - simply connected domain

Originally Posted by sinichko

Prove that if domain $\Omega$ conformaly maps to unit disc $D=\left \{ \left | z \right |<1 \right \}$then $\Omega$ it's simply connected domain.
You need to assume that the conformal map $f:\Omega\to D$ is bijective (and therefore invertible), otherwise the result is not true.

Suppose that S is a loop in $\Omega$. The image f(S) of S is a loop in the simply-connected space D. So there is a continuous transformation in D that contracts this loop to a point. The image in of that deformation under the map $f^{-1}$ will be a continuous transformation in $\Omega$ which contracts S to a point S to a point.

3. ## Re: Complex Analysis - simply connected domain

[QUOTE=Opalg;696240]You need to assume that the conformal map $f:\Omega\to D$ is bijective (and therefore invertible), otherwise the result is not true.

Thanks a lot!