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 $f^{-1}(U_{\alpha})$  is  a  good  cover for  $M$?