Could there exist two open, disjoint sets in $R^n$ s.t. their union is path connected?
I don't really know where to start with this, but for right now I think that trying to prove that for a set $U$, the set $U \cup U^c$ cannot be path connected. For any disjoint set $V$ , $V \subset U^c$, therefore $V \cup U$ cannot be path connected.
Is this correct? Thanks for your time.