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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.

Thank you very much for your helps!I want to modify my question.

My modified question:

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

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.

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

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.

Thank you very much for your helps!I want to modify my question.

My modified question:

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

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

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