 Solution

False. In $X=\mathbb{R}$, takeThen $A\cap B=\{2\}$ and $A\cup B=[0,2]$ are connected. Nevertheless $A$ is disconnected: $[0,1)$ and $\{2\}$ are disjoint nonempty relatively open sets partitioning $A$. For instance, intersecting $A$ with $(-1,3/2)$ and $(3/2,3)$ produces this separation. The endpoint missing from $[0,1)$ is supplied by $B$, so the union becomes connected without forcing $A$ itself to be connected.