The Topological Point of View

This book is about the topological approach to certain topics in analysis, but what does that really mean? Starting with the “epsilon-delta” parts of elementary calculus, analysis makes extensive use of topological ideas and techniques. Thus the issue is not whether analysis requires topology, but rather how central a role the topological material plays. Rather than attempt the hopeless task of defining precisely what I mean by the topological point of view in analysis, I’ll illustrate it by outlining two proofs of a well-known theorem about the existence of solutions to ordinary differential equations. In the first proof, the key step is the construction of a sequence of approximate solutions whose limit is the required solution. In the second proof, a general topological theorem about the behavior of selfmaps of linear spaces implies the existence of the solution. The two proofs have several features in common, including their dependence on a substantial topological result, but I trust that even my (intentionally) very sketchy treatment will make it clear how basic the differences are in the ways that the two arguments reach the same conclusion. Here’s the theorem.