For a manifold, $\mathcal{M}$ and a countable cover of it by coordinate domains $U_1, \dots, U_n \subset \mathbb{R}^N$ is there a way to "stitch" them together into a "common" Euclidean space with a common coordinate system?
It seems an intuitive enough "operation" but I am not at all sure of how to define a mechanism to do it. I was initially thinking of a map such as $F: U_1, \dots, U_n \rightarrow \mathbb{R}^N$ but I have never seen a map defined like this. Alternatively, would a definition such as $F: \mathbb{R}^N \rightarrow \mathbb{R}^N$ suffice or are there more requirements that need to be fulfilled here?