The Fundamental Group

Throughout this book, we have established different topological invariants such as connectedness and compactness as tools to classify topological spaces. In this last chapter, the concept of the fundamental group is defined. The idea is to consider the set of all loops with a fixed base point of a space, that is, continuous paths that begin and end at the same point, and give a structure of a group to this set. This group is called the fundamental group of the space and intuitively helps to detect the holes in a topological space. As another topological invariant, two homeomorphic spaces have isomorphic fundamental groups. The novelty here is that the problem of classification of topological spaces is now an algebraic question. This chapter gives an approach to the concept of the fundamental group, but it will serve us to show the power of using algebra in topology.