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.
1. Introduction 2. Notation and terminology 3. More on sets and functions Direct and inverse images Inverse functions 4. Review of some real analysis Real numbers Real sequences Limits of functions Continuity Examples of continuous functions 5. Metric spaces Motivation and definition Examples of metric spaces Results about continuous functions on metric spaces Bounded sets in metric spaces Open balls in metric spaces Open sets in metric spaces 6. More concepts in metric spaces Closed sets Closure Limit points Interior Boundary Convergence in metric spaces Equivalent metrics Review 7. Topological spaces Definition Examples 8. Continuity in topological spaces; bases Definition Homeomorphisms Bases 9. Some concepts in topological spaces 10. Subspaces and product spaces Subspaces Products Graphs Postscript on products 11. The Hausdorff condition Motivation Separation conditions 12. Connected spaces Motivation Connectedness Path-connectedness Comparison of definitions Connectedness and homeomorphisms 13. Compact spaces Motivation Definition of compactness Compactness of closed bounded intervals Properties of compact spaces Continuous maps on compact spaces Compactness of subspaces and products Compact subsets of Euclidean spaces Compactness and uniform continuity An inverse function theorem 14. Sequential compactness Sequential compactness for real numbers Sequential compactness for metric spaces 15. Quotient spaces and surfaces Motivation A formal approach The quotient topology Main property of quotients The circle The torus The real projective plane and the Klein bottle Cutting and pasting The shape of things to come 16. Uniform convergence Motivation Definition and examples Cauchy's criterion Uniform limits of sequences Generalizations 17. Complete metric spaces Definition and examples Banach's fixed point theorem Contraction mappings Applications of Banach's fixed point theorem Bibliography Index