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!