Let $X$ be a connected compact subset of the plane. Suppose that $A\cap B$ is connected for every two closed connected subsets $A,B\subseteq X$. Then is $X$ non-separating? That is, is $\mathbb R^2\setminus X$ connected?
It seems like the answer should be yes, and I have tried proving it but do not know how.
The answer is no. Take the two endpoint Knaster continuum and identify the two endpoints as shown in the image below:
[2-endpoint Knaster Continuum with Endpoints Identified][1]
The image is from my paper "Atriodic acyclic continua and class(W)" which appeared in Proceedings of the AMS, Vol. 90, No. 3, 1984, starting on p. 477.
Jim Davis [1]: https://i.stack.imgur.com/N16g3.png