Assume we have a bounded domain in the complex plane with smooth boundary. Is it possible to write the boundary as disjoint union of finitely/countably many closed (smooth) curves?
And furthermore, can you always obtain an “outer” closed curve whose interior contains the whole domain?
In general, if the boundary is not smooth, really strange things can happen. But is this case nice enough? I'm grateful for counterexamples or sketches of proofs.
By assumption $\partial U = \overline U - U$ is a smooth manifold (without boundary). Because $\partial U = \overline U \cap (\Bbb C - U)$, this is a closed set; you have assumed it is bounded, so it is compact. A compact connected curve without boundary is diffeomorphic to $S^1$, and a compact 1-manifold without boundary is diffeomorphic to some finite disjoint union thereof.
There is some difficulty in counting the number of components, which has to do with the topology of the domain. There is one of these per "end" of $U$. A simply connected domain has one end; an annulus has two. (In general, suppose $U$ is the interior of a compact manifold with boundary $\overline U$. Then the number of ends agrees with the number of boundary components, which furthermore agrees with the rank of $H_1(U)$ via what is called Poincare-Lefschetz duality.