This is a brief, clearly-written introduction to point-set topology. The approach is axiomatic and abstract — the development is motivated by a desire to generalize properties of the real numbers rather than a need to solve problems from other areas of mathematics. In particular there is very little mention of function spaces, although some of the examples deal with the existence of solutions to integral and differential equations as an application of Banach’s fixed-point theorem.
The book assumes some familiarity with the topological properties of the real line, in particular convergence and completeness. The level of abstraction moves up and down through the book, where we start with some real-number property and think of how to generalize it to metric spaces and sometimes further to general topological spaces. Most of the book deals with metric spaces.
The book has modest goals. It introduces the most important concepts of topology but does not take any of them very far. The exercises at the end of each chapter are partly routine applications of the chapter contents and partly extensions into more difficult areas not covered in the chapter. There is a companion web site that has solutions to all the exercises, as well as a great deal of supplemental material that did not fit into the main narrative. Because the book starts out with the real line, it is slanted somewhat towards analysis. Its aim is topology and it is not as nearly as thorough as analysis-oriented books such as Wilansky’s Topology for Analysis or Kelley’s General Topology. It also has an interesting chapter on quotient spaces, focused on Moebius strips and tori with various numbers of holes.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning.