Topology: Point-Set and Geometric (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)

The essentials of point-set topology, complete with motivation and numerous examplesTopology: 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 contextSections with exercise sets ranging in difficulty from easy to fairly challengingExercises that are very creative in their approaches and work well in a classroom settingA supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs This Text Covers The Essentials Of Point-set Topology In A Relatively Terse Presentation, With Lots Of Examples And Motivation Along The Way. Along With The Standard Point-set Topology Topics (connected Spaces, Compact Spaces, Separation Axioms, And Metric Spaces), The Author Includes Path-connectedness, And A Chapter On Constructing Spaces From Other Spaces (including Products, Quotients, Etc.). The Text Culminates In To Two Main Chapters, Each Independent Of The Other: 1) The Classification Theorem For Compact, Connected Surfaces And 2) Fundamental Groups And Covering Spaces, With Applications Giving The Reader The Choice Of Which Subject Best Suits Them. Foreword -- Acknowledgments -- 1. Introduction : Intuitive Topology -- 1.1. Introduction : Intuitive Topology -- 2. Background On Sets And Functions -- 2.1. Sets -- 2.2. Functions -- 2.3. Equivalence Relations -- 2.4. Induction -- 2.5. Cardinal Numbers -- 2.6. Groups -- 3. Topological Spaces -- 3.1. Introduction -- 3.2. Definitions And Examples -- 3.3. Basics On Open And Closed Sets -- 3.4. The Subspace Topology -- 3.5. Continuous Functions -- 4. More On Open And Closed Sets And Continuous Functions -- 4.1. Introduction -- 4.2. Basis For A Topology -- 4.3. Limit Points -- 4.4. Interior, Boundary And Closure -- 4.5. More On Continuity -- 5. New Spaces From Old -- 5.1. Introduction -- 5.2. Product Spaces -- 5.3. Infinite Product Spaces (optional) -- 5.4. Quotient Spaces -- 5.5. Unions And Wedges -- 6. Connected Spaces -- 6.1. Introduction -- 6.2. Definition, Examples And Properties -- 6.3. Connectedness In The Real Line -- 6.4. Path-connectedness -- 6.5. Connectedness Of Unions And Finite Products -- 6.6. Connnectedness Of Infinite Products (optional) -- 7. Compact Spaces -- 7.1. Introduction -- 7.2. Definition, Examples And Properties -- 7.3. Hausdorff Spaces And Compactness -- 7.4. Compactness In The Real Line -- 7.5. Compactness Of Products -- 7.6. Finite Intersection Property (optional). 8. Separation Axioms -- 8.1. Introduction -- 8.2. Definition And Examples -- 8.3. Regular And Normal Spaces -- 8.4. Separation Axioms And Compactness -- 9. Metric Spaces -- 9.1. Introduction -- 9.2. Definition And Examples -- 9.3. Properties Of Metric Spaces -- 9.4. Basics On Sequences -- 10. The Classification Of Surfaces -- 10.1. Introduction -- 10.2. Surfaces And Higher-dimensional Manifolds -- 10.3. Connected Sums Of Surfaces -- 10.4. The Classification Theorem -- 10.5. Triangulations Of Surfaces -- 10.6. Proof Of The Classification Theorem -- 10.7. Euler Characteristics And Uniqueness -- 11. Fundamental Groups And Covering Spaces -- 11. 1. Introduction -- 11.2. Homotopy Of Functions And Paths -- 11.3. An Operation On Paths -- 11.4. The Fundamental Group -- 11.5. Covering Spaces -- 11.6. Fundamental Group Of The Circle And Related Spaces -- 11.7. The Fundamental Groups Of Surfaces -- References -- Index. Paul L. Shick. Includes Bibliographical References (p. 263-264) And Index. The essentials of point-set topology, complete with motivation and numerous examples 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 The essentials of point-set topology, complete with motivation and numerous examples 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 topicsconnected and path-connected spaces, compact spaces, separation axioms, and metric spacesTopology 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: Topology is designed to be covered in one semester by typical math majors; however, it is still quite rigorous, modeling for the students how one writes precise proofs. It covers the essentials of point-set topology in a relatively terse presentation, with lots of examples and motivation along the way.