Let $\Omega \subset \mathbb R^2$ be a *bounded* $C^2$ domain. Is $\Omega$ then finitely connected? As I learned [recently](https://mathoverflow.net/questions/421658/what-is-a-finitely-connected-domain#comment1083842_421658bounded) a domain in $\mathbb R^2$ is finitely connected iff “[its] complement has finitely many components (holes)“.


The first (mostly smooth) bounded domain with infinitely many holes that comes to mind is
$$\Omega = B\_1(0) \setminus \overline{\bigcup\_{n=2}^\infty B\_{1/n^2}(1/n)},$$
however this domain fails to be $C^2$ at $0$. (Here $B\_{r}(x)$ is the ball with radius $r$ centered at $x$.)


Similar constructions should run into the same issue. In particular, suppose that $\Omega$ is not finitely connected and enumerate the holes by $A\_1, A\_2, \dots$. For $n \in \mathbb N$, take $x\_n \in \partial A\_n$. As $\partial \Omega$ is compact, there exists a converging subsequence. I doubt that $\Omega$ can be $C^2$ at this limit point. However, I fail to make this precise.


My main interest is to apply a result holding for finitely connected $C^2$ domains to a bounded domain and thus I would welcome a reference stating that bounded $C^2$ domains are always finitely connected. This is also the reason I require the domains to be $C^2$ but the same question can of course also be asked for Lipschitz domains, for instance.


**Edit:** A domain is $C^2$ if it can locally be written as the set of points above the graph of a $C^2$ function. A domain is bounded if it is contained in $B\_R(0)$ for some $R > 0$. A bounded $C^2$ domain is a domain which is both bounded and $C^2$.



Is every planar bounded $C^2$ domain finitely connected?
Yes. It is finitely connected. The boundary of your domain is a compact one-dimensional manifold and therefore it consists of a finite number of curves diffeomorphic to circles. You can find a proof of classification of compact one dimensional manifolds in:


**J.W. Milnor,** *Topology from the differentiable viewpoint.* Based on notes by David W. Weaver University Press of Virginia, Charlottesville, Va. 1965.


