What does "relatively closed" mean?
Solution 1:
This should mean that $S$ is a closed subset of the topological space $U$, where the topology on $U$ is the subspace topology it gains as a subset of $\mathbf R^n$. Explicitly, this means that there is a closed subset $\widetilde S$ of $\mathbf R^n$ such that $S = U \cap \widetilde S$. As Shawn notes in the comments, a good example is the relatively closed subset $[1/2, 1)$ of the interval $(0, 1)$.
Note that $U - S$ is also open and closed in $U$. If $U - S$ were nonempty, then $U = S \cup (U - S)$ would contradict the connectedness of $U$.