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For a senior undergraduate or first year graduate-level course in Introduction to Topology. Appropriate for a one-semester course on both general and algebraic topology or separate courses treating each topic separately.
This text is designed to provide instructors with a convenient single text resource for bridging between general and algebraic topology courses. Two separate, distinct sections (one on general, point set topology, the other on algebraic topology) are each suitable for a one-semester course and are based around the same set of basic, core topics. Optional, independent topics and applications can be studied and developed in depth depending on course needs and preferences.
I. GENERAL TOPOLOGY.1. Set Theory and Logic.
2. Topological Spaces and Continuous Functions.
3. Connectedness and Compactness.
4. Countability and Separation Axioms.
5. The Tychonoff Theorem.
6. Metrization Theorems and Paracompactness.
7. Complete Metric Spaces and Function Spaces.
8. Baire Spaces and Dimension Theory.
II. ALGEBRAIC TOPOLOGY.9. The Fundamental Group.
10. Separation Theorems in the Plane.
11. The Seifert-van Kampen Theorem.
12. Classification of Surfaces.
13. Classification of Covering Spaces.
14. Applications to Group Theory.
- NEW - Greatly expanded, full-semester coverage of algebraic topology—Extensive treatment of the fundamental group and covering spaces. What follows is a wealth of applications—to the topology of the plane (including the Jordan curve theorem), to the classification of compact surfaces, and to the classification of covering spaces. A final chapter provides an application to group theory itself.
Follows the present-day trend in the teaching of topology which explores the subject much more extensively with one semester devoted to general topology and a second to algebraic topology. Ex.___
- Advanced topics—Such as metrization and imbedding theorems, function spaces, and dimension theory are covered after connectedness and compactness.
- Order of topics proceeds naturally from the familiar to the unfamiliar—Begins with the familiar set theory, moves on to a thorough and careful treatment of topological spaces, then explores connectedness and compactness (with their many ties to calculus and analysis), and then branches out to the new and different topics mentioned above.
Carefully guides students through transitions to more advanced topics being careful not to overwhelm them. Motivates students to continue into more challenging areas. Ex.___
- One-or two-semester coverage—Provides separate, distinct sections on general topology and algebraic topology.
Each of the text's two parts is suitable for a one-semester course, giving instructors a convenient single text resource for bridging between the courses. The text can also be used where algebraic topology is studied only briefly at the end of a single-semester course. Ex.___
- Many examples and figures—Exploits six basic counterexamples repeatedly.
Avoids overemphasis on “weird counterexamples.” Ex.___
- Exercises—Varied in difficulty from the routine to the challenging. Supplementary exercises at the end of several chapters explore additional topics.
Deepen students' understanding of concepts and theorems just presented rather than simply test comprehension. The supplementary exercises can be used by students as a foundation for an independent research project or paper. Ex.___