Let $M$ be a (not necessarily compact)) smooth manifold.

1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U_{\alpha}$ such that each $f^{-1}(U_{\alpha})$ is homeomorphic to $\mathbb{R}^{n}$?

2.Is there a smooth map $f:M \to \mathbb{R}^{k}$, for some $k \in \mathbb{N}$ and an open covering $\mathbb{R}^{k}=\cup U_{\alpha}$ such that $f^{-1}(U_{\alpha}),s$ is a good cover for $M$?