Introduction to Topology: Pure and Applied

This rather unusual book aims to introduce topology as a mathematical discipline and to demonstrate the value of topological ideas in various areas of science, engineering and mathematics. It is intended to serve as a text for a one or two semester undergraduate course for students who have completed an introductory course in abstract mathematics or for mathematically mature students who have taken the basic calculus sequence.

A reader of this text will learn the basics of point set topology in the first seven chapters, and then be introduced to a variety of topics including knots, manifolds, dynamical systems, fixed point theorems, and graph theory. The authors regard point set topology as the core of the book. For each of the other topics they offer a introductions designed to elicit student interest and facilitate further investigation. Applications are sprinkled liberally throughout; they are well-selected and germane to real problems.

The authors’ approach to point set topology has a good deal of pedagogical merit. Point set topology can be rather a dry subject, but a mixture of applications and good illustrations here keep it lively. For example, digital topology (useful in digital image processing) and phenotype spaces (of value to molecular biologists) are considered just after the concept of topological space is introduced. Geographic Information Systems are introduced after an initial discussion of open and closed sets, boundaries and limit points. Discussions of the applications here are not particularly deep, but they help motivate students and broaden their thinking.

The treatment of point set topology is neither as complete nor as extensive as in more typical textbooks. So, for example, infinite product spaces and their complications are not considered, the separation axioms get a much abbreviated treatment, and there’s not much of the usual zoo of pathological examples. This seems eminently sensible to me; students needing more specialized background will have a more than adequate basis to build upon.

The mixture of topics in the remaining chapters gives the instructor a lot of options. One could spend the better part of a term on homotopy and degree theory, followed by fixed point theorems and applications to economics and game theory. Alternatively, one could concentrate on topological graph theory and emphasize applications to chemistry. The final chapter on manifolds and cosmology addresses the fascinating question of the shape of the universe and what the cosmic microwave background might tell us about it.

The book has plenty of exercises, many of them standard. A notable feature is an excellent set of references to applications of topology in seventeen different areas.

A minor complaint I have is that the pages in the book are over-full. I expect this is a consequence of trying to keep down the page count and cost, but many pages look too crowded. It also might have been desirable to add a chapter on homology since there are many new and exciting applications there.

Overall, this is a creative and valuable addition to the world of introductory topology textbooks.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.