I'm studying for an exam which is a couple months away and I found an old exam which asks the following: Find the fundamental group of: a) The closed subset in R3 given by the equation x - y^2 -z^2 = in the standard coordinates. b) The closed subset in R3 given by the equation x - y^2 -z^2 + 1 = in the standard coordinates. c) The one point compactifcation of the disjoint union of two open discs in R^2 d) The space of upper triangular matrices of the size 2 by 2 over C (complex) with derminant equal to 1 considered as a closed subspace in the vector space of all matrices of the size 2 by 2 over C (complex). Explain. Now,my book (as well as Wikipedia) both define the fundamental group similarly. That is: fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. What I want to know is how to solve the above? In my book, there are no examples of such things. I want to know if there is a step by step process for each, or if each case is completely different. Thank you for your time.