Let $C \subset X$ be a connected space and $A \subset X$ s.t. $C\cap A \neq \emptyset$ and $C\cap (X-A) \neq \emptyset$. Prove... Let $C \subset X$ be a connected space and $A \subset X$ s.t. $C\cap A \neq \emptyset$ and $C\cap (X-A) \neq \emptyset$. Prove that $C\cap \partial A \neq \emptyset$.
I don't know how to prove this since I fail to see how I can use the fact that C is connected. Any hint would be appreciated.

That $C$ is connected means it is not a union of two disjoint sets that are open in $C$. Note that $C \cap \mathring A$ and $C \cap \mathring {X\setminus A}$ are both open in $C$.
