Basically the title: Under what conditions can I select a neighbourhood $U$ of $\overline{\Omega}$ such that there exists a continuous projection $U\to\overline{\Omega}$? Here, $\Omega$ is a bounded domain (i.e. open and connected) in $\mathbb{R}^n$ and I would also like the projection to correspond with the identity map when restricted to $\overline{\Omega}$.
I was considering, for a neighbourhood $U$ of $\overline{\Omega}$, the projection map $\pi : U\to\overline{\Omega}$ taking a point $p\in U$ to it's nearest neighbour $\pi(p)\in\overline{\Omega}$. Clearly, $\pi(\omega) = \omega$ for any $\omega\in\Omega$ as desired.
If $\Omega$ has continuous boundary and I choose $U$ sufficiently "small", can I conclude that $\pi$ is continuous? Do I need a stronger assumption on $\partial\Omega$?
I appreciate any help/comments. Thank you in advance.