# A bounded, open, connected set that contains infinitely many open sets but not their boundaries?

Let $V$ be a bounded, open, connected set of $\mathbb{R}^{n}$ ($n>1$). Does there exist an infinite sequence of disjoint open sets $(C_{n})$, all included in the closure of $V$ and such that the boundary of each $C_{n}$ is included in the boundary of $\overline{V}$?

Note: This is a very similar question to A bounded open set that contains infinitely many open sets but not their boundaries? but with one extra interesting condition.

If $V$ is connected, so is $\overline{V}$, and also $U = \overset{\Large\circ}{\overline{V}}$, since $V \subset U \subset \overline{V}$. We thus may assume that $V = U$. Consequently, $\partial \overline{V} = \partial V$.
If $W \subset \mathbb{R}^n$ is open with $W \subset \overline{V}$, then it follows that $W \subset V$. If also $\partial W \subset \partial V$, then $W$ is open and closed in $V$. Since $V$ is connected, it follows that either $W = \varnothing$ or $W = V$.
Thus, unless one allows a sequence containing infinitely many terms $C_k = \varnothing$ and at most one term $C_m = V$, no such sequence exists.