In my complex analysis class we’re currently covering the Riemann Mapping Theorem, and as is well-known there are no conditions imposed on the actual shape of the region $\Omega$, except that it must be open, simply connected, and not all of $\mathbb{C}$. So I was wondering how pathological such a region can really be, and particularly its boundary $\partial \Omega$. For example, can the boundary be a non-nullset? No-where differentiable? Can it be that the interior of the closure of $\Omega$ is not the same as $\Omega$ itself? And so on.
This question may also be extended to $\mathbb{R}^n$. However, I feel like “simply connected” is no longer the right analogue here, because $\mathbb{R}^3$ with a disjoint set of points removed is simply connected, which is not true of $\mathbb{R}^2$. I think to generalize, the question would have to be “How pathological can the boundary of an open, connected subset $\Omega \subset \mathbb{R}^n$ be, given that $\mathbb{R}^n \cup \{\infty\} \setminus \Omega$ is also connected?”.
Let $E \subset [0,1]$ be a "fat Cantor set": closed, nowhere dense, with positive Lebesgue measure. Then $((-1,1) \times (-1,2)) \backslash ([0,1] \times E) $ is open in $\mathbb R^2$ and simply connected, and its boundary contains $[0,1] \times E$ which has positive two-dimensional Lebesgue measure.