A space X is unicoherent if whenever A,B are closed connected subsets of X such that , their intersection is connected. The survey "A Survey on Unicoherence and Related Properties" by Garcia-Maynez and Illanes says that intuitively, a unicoherent space is a space with no "holes". For example, the closed disk in the plane is unicoherent, but the circle is not. This sounds like simple connectedness. For simple connectedness, by van Kampen's theorem, we have the following: if a space X has simply connected open subsets U,V such that and is nonempty and path-connected, then X is simply connected. It is then natural to ask a similar question for unicoherence. So, my question is: if a space X has closed unicoherent subsets A,B such that and is connected, is X necessarily unicoherent? If necessary, you can assume that A,B are open rather than closed, or replace unicoherence conditions with open unicoherence conditions. Also, you can assume that is nonempty if needed. Thank you in advance!