Say $X$ is a non-connected topological space, i.e. $X= U\cup V$, and $U,V$ are disjoint (non-empty) open sets. Then suppose $C$ is a connected subspace of $X$, with the standard subspace topology. Can $C$ be in $U$ and in $V$?
I think the answer is no, because $U\cap C, V\cap C$ would be a seperation of $C$. Is this correct?
The answer is no for the precise reason you state.