Guillemin and Pollack define manifolds to be subsets of $\mathbb R^n$ locally diffeomorphic to open subsets of $\mathbb R^k$, and they define the notion of a smooth map on (not necessarily) open subset of $\mathbb R^n$ by saying that it is smooth if it locally extends to smooth maps on open subsets of $\mathbb R^n$. This definition works for manifolds (as they define it), so charts are not used as far as I can see.
In the standard definition of a manifold, the notion of a smooth map between manifolds is defined in terms of charts.
So I have two questions:
- First, if the fact that a manifold in the sense of Guillemin-Pollack is locally diffeomorphic to $\mathbb R^k$ is not used for defining smooth maps, what for do we need it? Only to tell the dimension of a manifold? Wouldn't it be sufficient then to require that a manifold must be locally homeomorphic to $\mathbb R^k$?
- If I define smooth maps between manifolds in the sense of Guillemin-Pollack in the usual fashion through charts, will it be equivalent definition?