I'm just trying to understand something about manifolds. What is meant when a manifold doesn't have boundary? I thought the boundary was where the manifold "ends" so to speak. Like a boundary point, something where you take a small nhbd (neighborhood) and you get something inside the set and then something outside the set. But then there are objects that arn't manifolds apparently but they do have boundary. Another thing. Excuse the simplicity of this question, but why are we so concerned about making maps and charts and atlases and such. What is the whole purpose of doing this? And why do we care about these maps being smooth? Also what is the purpose of restricting things like here with local diffeomorphisms. http://en.wikipedia.org/wiki/Local_diffeomorphism As seen in the link they restrict F which is the map to U. Why do this? What exactly do they mean by this? Thanks.