What type of functions map a simply connected set onto a simply connected set

Question

As the question title states, in what cases can I say that a bijective continuous function $f(z): \mathbb{C}\to \mathbb{C}$ maps all simply connected sets $A\in\mathbb{C}$ to a simply connected set $B = \{f(x), x\in A\}$?

Answer 1

Every continuous bijection $f:\mathbb{C}\to\mathbb{C}$ is a homeomorphism, by invariance of domain. So, for any $A\subseteq \mathbb{C}$, $f$ restricts to a homeomorphism $A\to f(A)$, and in particular $A$ is simply connected iff $f(A)$ is simply connected.