Foundations of Topology

Originally published in 1993, Foundations of Topology was re-issued with a new publisher this year. I haven’t tracked down a copy of the original edition, so I don’t know how this compares. I do know that this edition provides a first course in point set topology with the usual touch of algebraic topology: the fundamental group and covering spaces.

The text is nicely written. But there are several nicely written books at this level suitable for an undergraduate course and/or a first semester graduate course. How does this one compare? What makes this one different?

Patty’s book mirrors the table of contents of Munkres’ classic text Topology: A First Course pretty well. Other books which aim for the undergraduate level often don’t include anything which could be called “Applications of Homotopy” but Patty has a whole chapter which provides a topological proof of the Fundamental Theorem of Algebra and a proof of the Jordan Curve Theorem. (These are also part of Munkres’ last chapter).

The inclusion of the special topics chapter is the difference I see. Patty sets up the Implicit Function Theorem and the Inverse Function Theorem for Banach spaces. To do this he discusses normed linear spaces and the Fréchet derivative. He provides enough to state the classification of compact surfaces too. This is just one section and supplies only references to a proof, but its inclusion does allow the student a glimpse and provides a jumping off point for learning more about geometric topology just as his discussion of the implicit and inverse function theorems allows a glimpse into the world of differential topology.

This is a book I would recommend to students. And, depending on the particular students and my particular focus in a given year, is one I would certainly consider using in my undergraduate topology course. While Adams and Franzosa (Introduction to Topology: Pure and Applied) do a better job of applying topology and Shick’s text (Topology: Point-Set and Geometric) is written with a bit more discussion and includes a full treatment of the classification of surfaces, Patty’s text provides all the necessities — including many exercises — and includes springboard material for a variety of special projects/presentations.

Michele Intermont is an associate professor of mathematics at Kalamazoo College. When she’s not doing mathematics, she’s swimming!