Universal covering space of connected open subset of $\mathbb R^n$

Question

Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?

Answer 1

No. For example, let $n=3$ and $U=\Bbb{R}^3\setminus \{(0,0,0)\}$. Then the universal cover of $U$ is $U$ itself (since it's simply connected) which is not homoemorphic to $\Bbb{R}^3$ (since it has nontrivial $2$-homology say).