Topology is an extremely important and fascinating field within mathematics. In the course of studying this subject, you not only encounter new concepts and methods, but also find them tying back into things you’ve already learned, such as continuous functions. But that alone isn’t enough to convey how important topology is — its real importance lies in the fact that it has a clear influence on almost every branch of mathematics. If you want to become a mathematician, whether your interest lies in algebra, analysis, operations research, or statistics, topology will be relevant to all of them. In modern mathematics, certain concepts from topology — compactness, connectedness, denseness — are as fundamental as sets and functions are in mathematics generally.

Topology has many different branches: general topology (also called point-set topology), algebraic topology, differential topology, and topological algebra. General topology is the introductory course for the field. In this book, the basic concepts of general topology will be explained in detail.

If you haven’t previously studied an axiomatic mathematical discipline like abstract algebra, learning how to construct proofs can be difficult. So, to help you learn how to prove things, the proofs in the early chapters will include some “asides” — these asides aren’t part of the proof itself, but they sketch out why the proof proceeds the way it does and how the underlying idea arose.

The book contains a great many exercises, and only by working through a large number of them can you really master this subject. Solutions to the exercises are not provided at the back of the book — though this is an unpopular choice, I’m sticking with it. The book already contains enough examples and proofs to help you work out the problems yourself, so there’s no need to provide additional solutions. New concepts are often introduced within the exercises themselves; generally speaking, I’ll bring up again later whatever concepts I consider important.

Finally, I should note one thing: the best way to understand why a given piece of mathematics came about is to read the history of mathematics. Unfortunately, this book doesn’t go into that in full. Appendix 2 offers some excerpts introducing famous figures in the history of topology, most of them selected from “The MacTutor History of Mathematics Archive.” Readers would do well to visit the website itself to read the complete articles and learn about other important figures. Bear in mind that understanding history through a single source is never enough.

Translated from Topology Without Tears / Sidney A. Morris [Oct. 14, 2007]