Here is a sampling of problems on differential topology. These problems are for your benefit, to make sure that you are comfortable with the basics of manifold theory. You do not need to turn them in, although you are welcome to if you want me to check your answers. do Carmo, chapter 0: # 2, 4, 9, 12 Spivak, volume I, chapter 1: # 1, 9, 11 chapter 2: # 3, 28, 33 chapter 3: # 7, 12, 19 Also, some of you might be interested in looking at Theorem 1 of appendix A of Spivak, where he proves that a manifold is second countable if and only if it is paracompact.