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 $U_1$ is disconnected. Then $U_1 = V \cup W$ where $V$ and $W$ are non-empty, disjoint, open subsets of $X$. Since $U_1 \cap U_2$ is connected, one of $V \cap U_2$ and $W \cap U_2$ must be empty. Suppose without loss of generality that $V \cap U_2 = \varnothing$. We have $U_1 \cup U_2 = V \cup W \cup U_2$. Note that $V$ and $W \cup U_2$ are non-empty and open. Furthermore, $V \cap (W \cup U_2) = \varnothing$. It follows that $U_1 \cup U_2$ is disconnected.
