Let $U,V$ be two nonempty connected open sets in $\mathbb{R}^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty connected open set.



a question about connected open sets in $R^2$
Let $V$ be the complement of a point $a$, and $U$ the complement of a ray $r$ with end-point $a$. Every ball not containing $a$ and intersecting $r$ is in fact split by $r$.


