In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary.
My attempt: Since both my sets need to be bounded, my thought was to limit them by the unitary ball. Its easy to construct two disjoint open sets that have a common boundary in the interior of the unitary circle, the problem is, of course, the exterior (the boundary of the union). So I considered two closed paths, helix-like curves that in polar coordinates can be expressed by
$\gamma(t) = (1-e^{-t},t)$
$\gamma_2(t) = (1-e^{-0.5t},t)$
So, set $U$ consists of the points between the two curves, that is, $U = \{(\rho, \theta) : 1-e^{-0.5\theta}<\rho<1-e^{-\theta}\}$ and $V = B_1(0)\setminus \overline{U}$. It's easy to show that $U$ is open, bounded and since it's path-connected it is connected. Same goes to $V$ and I guess $\partial U = \partial V = Im(\gamma)\cup Im(\gamma_2) \cup D_1$ ($D_1$ being the unitary disk).
My questions: Is there an easier construction? Is my construction right?