Topology: Point-Set and Geometric
Topology: Point-Set and Geometric presents an introduction to topology that begins with the axiomatic definition of a topology on a set, rather than starting with metric spaces or the topology of subsets of Rn. This approach includes many more examples, allowing students to develop more sophisticated intuition and enabling them to learn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors.
Along with the standard point-set topology topics—connected and path-connected spaces, compact spaces, separation axioms, and metric spaces—Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students.
Topology: Point-Set and Geometric features:
- A short introduction in each chapter designed to motivate the ideas and place them into an appropriate context
- Sections with exercise sets ranging in difficulty from easy to fairly challenging
- Exercises that are very creative in their approaches and work well in a classroom setting
- A supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs
1. Introduction: Intuitive Topology.
2. Background on Sets and Functions.
3. Topological Spaces.
4. More on Open and Closed Sets and Continuous Functions.
5. New Spaces from Old.
6. Connected Spaces.
7. Compact Spaces.
8. Separation Axioms.
9. Metric Spaces.
10. The Classification of Surfaces.
11. Fundamental Groups and Covering Spaces.
The book has been thoroughly class tested for three semesters and has received very positive feedback from students.
Each chapter has a short introduction designed to motivate the ideas and place them into an appropriate context.
Each section has a set of exercises, ranging in difficulty from easy to fairly challenging. The exercises are very created in their approach and work well in a classroom setting.
The text is supported by a related website that contains more complete (and colorful) illustrations of certain objects, several learning modules that illustrate complicated topics, and some animations of particularly complex proofs.