TITLE: a question about connected open sets in $R^2$
QUESTION [10 upvotes]: 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.

REPLY [13 votes]: 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 $r$.

REPLY [6 votes]: Choose an enumeration $\{ r_n \}_{n \in \mathbb{Z}_{\geq 0}}$ of the rational numbers, and form the subset $A \subset \mathbb{R}$ given by the union of neighborhoods of radius $2^{-n}$ around $r_n$.  Let $U$ be the union of $A \times \mathbb{R}$ with a suitable open half-plane that makes the set connected.  Any ball $B$ that is both disjoint from the half-plane and not contained in $U$ satisfies the property that $B \cap U$ has infinitely many connected components.  Thus, we may choose $V$ to be the union of $U$ and the complement of $A \times \{0\}$.