Let $S$ be a triangulation of a closed surface and $T \subseteq S$ its subcomplex. Let $C$ be a connected component of the topological space $|S| - |T|$ (here, $|S|$ denotes the polytope of the simplicial complex $S$). Suppose that $|T|$ is connected. I'm trying to prove that the set-theoretic boundary of $C$ is a connected graph. It's easy enough to prove that the boundary of $C$ is a $1$-dimensional simplicial complex, but I don't know how to prove (or disprove) that it is connected. Any hint to a proof or a counterexample would be much appreciated.
Thanks!