analysis - Is the support of a compactly supported function on $Omega$ a proper subset of $Omega$?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$ be open. Is there some continuous $\phi:\Omega\to\mathbb R$ with compact support $\operatorname{supp}$ and $\operatorname{supp}\phi=\overline{\Omega}$?