If $U$ and $U'$ be two domains in $\Bbb C$, and $f$ be a homeomorphism in $U$ and $U'$ then domain $U$ is simply connected $\iff$ $U'$ is simply connected. I found this problem in complex analysis. So I would prefer to know its proof from complex point of view rather using topological propositions. Thanks.
There are few properties which are equivalent for a domain $D$ in Complex plane.
a)$D$ is simply connected.
b)for each $z_0\in \Bbb C$\ $D$ there is a analytic branch of $log(z-z_0)$ defined on $D$.
c)The compliment of $D$ in the extended complex plane $\Bbb C^*$ is connected.