The first part of this book stresses broad concepts and examples, the second rigorous theory. Topology allows us to make the transition. Why does topology play this critical role? Essentially because topology allows us to make headway in characterizing qualitative properties of a model when quantitative information is missing. Why not seek better quantitative information instead? This goes to the heart of the matter, to the basic question of what competitive theory is really about. In modeling a competitive economy we want to assume as little as possible about what consumers and firms are like because it is the absence of such information which renders planners impotent and markets worth having. At least that is what we, as economists, have claimed from Adam Smith on. Modern general equilibrium theory can be faulted for not going far enough in this direction, but it certainly does try.

This chapter looks at individual consumers and firms in isolation, ignoring for now how they interact with one another. The main issues concern their response to prices and other aspects of their environment. The questions we ask are deliberately qualitative, not quantitative. Not what formula describes the demand of a consumer with Cobb-Douglas utility, but rather: Does the consumer have a best response to her environment? If the best response is not unique, what more can we say about the set of best responses? Does the response vary continuously when the environment changes? Without topology we cannot get far in addressing these questions. With topology we can say quite a lot.