Extending a connected open set Assume $\emptyset\neq V\subseteq U\subseteq\mathbb{R}^n$ are open and connected sets so that $U\setminus\overline{V}$ is connected as well. Given any point $x\in U$, is there always a connected open set $W\subseteq U$ so that $\{x\}\cup V\subseteq W$ and $U\setminus\overline{W}$ is connected? In other words, can $V$ be extended to a connected open set containing a given point so that the complement of the closure of the extended set is still connected?

Note that it is enough to show this only for $x ∈ ∂V ∩ U$. Consider $A := \{x ∈ U: ∃W ⊆ U \text{ connected, } \{x\} ∪ V ⊆ W, U \setminus \overline{W} \text{ connected}\}$ This is obviously open subset of $U$. If it is also closed then it's whole $U$.
