Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many smooth arcs. Consider a disk $ D $ centered at a point $x \in \Omega $ and denote by $ \Omega\_0 $ the connected component of $ \Omega \cap D $ that contains $ x $.


I wish to relate the number of connected components of $ \partial \Omega $ to the number of connected components of $ \partial \Omega\_0 $. Is it true that the first is not less than the latter?


Any comments and references are welcome. Thanks!



Boundary components of a subsurface
(Based solely on the topology:)


Since $\Omega\_0$ is connected and contained within the disc $D$, it must be a planar surface, ie. a disc with holes. It has one 'outer' boundary component that is made up at least in part of $\partial D$ (unless $\partial D$ is disjoint from $\Omega$, in which case $\Omega\_0=\Omega$), and all other boundary components of $\Omega\_0$ are complete boundary components of $\Omega$. Thus $\Omega\_0$ could have one more boundary component than $\Omega$ (when $\partial D$ is contained in the interior of $\Omega$) and otherwise it has at most as many as $\Omega$.


