TITLE: The image of simply-connected domain
QUESTION [14 upvotes]: If $\Omega$ is a simply-connected domain in $\mathbb R^n$ and $f$ is a injective continuous map from $\Omega$ to $\mathbb R^n$, then is it necessary that $f(\Omega)$ a simply-connected domain?

REPLY [4 votes]: It is worth noting that a very slight modification, where we allow the dimension of the ambient space to be different, makes the answer "no".
If $\Omega$ is a simply-connected domain in $\mathbb{R}^n$ and $f$ is an injective continuous map from $\Omega$ to $\mathbb{R}^m$, then $f(\Omega)$ is not necessarily a simply-connected domain.
An example is given by the "figure 6", thought of as a map from an open interval in $\mathbb{R}^1$ into $\mathbb{R}^2$.