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

Suppose $A$ and $B$ form a separation of $U_1$, i.e., $A$ and $B$ are disjoint nonempty open sets such that $A\cup B = U_1$. Because $U_1 \cap U_2$ is a connected subset of $U_1$, it must be entirely contained in either $A$ or $B$ (else we would get a separation of $U_1 \cap U_2$ by intersecting $A$ and $B$ with $U_1 \cap U_2$); WLOG let $U_1 \cap U_2 \subset A$. Then, $A\cup U_2$ and $B$ forms a separation of $U_1 \cup U_2$, a contradiction.
