 Solution

True. Suppose $A\cup B=U\bigsqcup V$ were a separation into nonempty relatively open sets. Intersecting with $A$ gives a separation unless all of the connected set $A$ lies on one side; similarly all of $B$ lies on one side. A point of $A\cap B$ forces both to lie on the same side. The other side would then be empty, a contradiction. Hence $A\cup B$ is connected.