Is an open connected subset of Euclidean space a countable sum of open precompact connected subsets?

Question

Let $U$ be an open subset in $\mathbb R^n$. Then there exists a sequence $(U_n)_{n=1}^\infty$ of open precompact subsets of $\mathbb R^n$ such that $U_n \subset cl U_{n+1} \subset U$ and $\bigcup_{n=1}^\infty U_n=U$.

Let's assume that additionally $U$ is connected. Does there exists
a sequence $(U_n)_{n=1}^\infty$ of open precompact subsets of $\mathbb R^n$ such that $U_n \subset cl U_{n+1} \subset U$ and $\bigcup_{n=1}^\infty U_n=U$ and additionally each $U_n$ is connected?

Thanks

Answer 1

Sure. Fix a point $x_0\in U$. Let $U_n$ be the set of points reachable from $x_0$ by a path lying in $U$ having length $<n$ and keeping a distance $>1/n$ from the complement of $U$.