Is the following set connected given that the union and intersection is connected Suppose $U_1, U_2$ are open sets in a space $X$. Suppose $U_1 \cap U_2$ and $U_1 \cup U_2$ are connected. Can we conclude that $U_1$ must be connected??
I am trying to find a counterexample, but I failed. PErhaps it is true? Can someone help me find a counter example? thanks

We assume that both $U_1$ and $U_2$ are nonempty (otherwise the result is trivial). If $U_1=A\cup B$, being $A$ and $B$ disjoint open sets, then $U_1\cap U_2$ is the disjoint union of the open sets $A\cap U_2$ and $B\cap U_2$, so by connectedness one of the sets is empty, say $A\cap U_2=\emptyset$. Therefore $U_1\cup U_2$ is the disjoint union of the open disjoint sets $A$ and $B\cup U_2$, so again by connectedness one of these sets is empty, namely $A$ (because $U_2\ne\emptyset$ by assumption). This proves that $U_1$ is connected.
