2.3 - Innovative Possibilities for Undergraduate Topology

Summary

Introduction

The development of topology ranks as one of the great success stories of twentieth century mathematics. While the precise definition of a topological space is not yet a full century old, the subject has become a core requirement for many branches of current mathematics research. From genetics to string theory to the social sciences, applications of topology are diverse and pervasive. In its own right, topology is a vital and ever growing area, comprising dozens of subfields and engaging hundreds of researchers around the world.

The status of the undergraduate semester-course in topology is, unfortunately, not quite so glorious. Introductory topology tends to be viewed as a course suitable primarily for students headed to graduate school. While there are many superb textbooks in the field, most pitch the subject at an advanced level, including far more material than is possible to cover in one semester. Ironically, the axiomatic rigor that makes topology a model and solid foundation for other fields is precisely the characteristic that makes it a difficult fit for the undergraduate curriculum.

In this paper, I hope to indicate how an introductory topology course can become an accessible and popular elective for math majors of various strengths and diverse goals. One of the great advantages of topology is the almost visual elegance of its formalism. By emphasizing this quality, a teacher can help students cope with the level of abstraction that is endemic to all theory courses.