Hi, everyone: I am a little confused about the issue of the inclusion map on submanifolds. AFAIK, if S is a submanifold of M , then if we give S the weak (or initial) topology of the inclusion, then the inclusion map is a homeomorphism.(his is the way I understand, of making the inclusion map of a subspace into a space, into a homeomorphism. If S is open in M, with this initial topology, then I think inclusion is also a diffeomorphism.) Question: under what conditions on S is the inclusion map a diffeomorphism?. I think that if S is closed in the topology of M, or at least not open in M , and given the weak topology, then the inclusion may not be a diffeomorphism. Is this correct?. And when is a (topological) subspace S of M a submanifold, other than when S is open in M, i.e., S is open as a subset of M? Thanks.