Construction of a diffeomorphism

Question

Let $M$ be an arbitrary smooth manifold with dimenison $n$, and $\{U_i\}$ be an open covering of $M$.

Can we construct a diffeomorphism $f_i:U_i \to B_i$?, where $B_i$ is the open unit ball in $\mathbb{R}^n$.

Answer 1

If $(U_i)$ is an arbitrary open covering, then no. Think of the case where $M$ is an open subset of $\mathbb R^n$, with covering just itself. Is it true that any open subset of $\mathbb R^n$ is diffeomorphic to $B_i$?