Let $U \subset \mathbb{R}^{m}$ be a open subset. Does there exists $f \in C_c^\infty(\mathbb{R}^{m})$ such that $U=f^{-1}\{(0,\infty)\}$?

Question

Let $U \subset \mathbb{R}^{m}$ be a open subset. Does there exists $f \in C_c^\infty(\mathbb{R}^{m})$ such that $U=f^{-1}\{(0,\infty)\}$?

Let $F=\mathbb{R}^{m}\setminus U$. I tried to use the results related to the partition of unity to construct (explicitly) a smooth function such that $f^{-1}(\{0\})=F$.

However, through this construction I can't guarantee that this function has compact support.