In the attached pdf file i have a few questions on manifolds, I hope you can be of aid. I need help on question 1,2,6,7. here's what I think of them: 1. a) the definition of a smooth manifold is that for every point in M we may find a neighbourhood W in R^k of it which the intersection W with M is diffeomorphic to an open neighbourhood U in R^m. Now if we take a connected component of M, say V, if it intersects W then the restricition of the above diffeomorphism will do. what do you think of this? b) I don't think it's irrelevant that M is even a manifold, cause if M is compact and its connected components are disjoint subsets which are connected which their union is M, then obviously if we take some open covering of M then by compactness there's a finite covering of M, this also covers each of its componets, or we may assume that each component has some covering and unite them, it's obviously a covering of M and thus by compactness have a finite covering. 2.I think we need to show this inductively, or better way, if we define M=U(MnW_i) where the union runs through i, MnW_j is the intersection of M with W_j, such MnW_j is diffeomorphic to some open neighbourhood in R^m. now we may increase the W_j as big as we please, and thus get increasing sets and if we take the closures of W_j's then those still cover M and are compact in M. not sure if that will work though. on questions 6 and 7 I'll ask later perhaps tommorrow or the day after that.