Hi, everyone: I have been reading Boothby's intro to diff. manifolds, and in def. 4.3, talking about 1-1 immersions F:N->M , n and m-mfld. and M an m-mfld. That: "...F establishes a 1-1 correspondence between N and the image N'=F(N) of M. If we use this correspondence to give N' a topology and a C^oo structure, then N' will be called a submanifold, and F:N->N' a diffeomorphism". I am just wondering how this topology and C^oo structure are defined. I imagine we are using pullbacks here, tho, since F is not necessarily a diffeom. , we cannot pullback by F, except maybe locally, using the inverse fn. theorem. And, re the topology on F'(N) , we cannot use subspace, since F may not be an embedding. Maybe we are using quotient topology, but it does not seem clear. Anyone know how the topology and C^oo structure are defined in N'=F(N)? I would appreciate any help.