Topology Through Inquiry

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Topology Through Inquiry

Michael Starbird : University of Texas at Austin, Austin, TX

Francis Su : Harvey Mudd College, Claremont, CA

MAA Press: An Imprint of the American Mathematical Society

Softcover ISBN:	978-1-4704-6261-1
Product Code:	TEXT/58.S

List Price:	$75.00
MAA Member Price:	$56.25
AMS Member Price:	$56.25

eBook ISBN:	978-1-4704-5532-3
Product Code:	TEXT/58.E

List Price:	$69.00
MAA Member Price:	$51.75
AMS Member Price:	$51.75

Softcover ISBN:	978-1-4704-6261-1
eBook: ISBN:	978-1-4704-5532-3
Product Code:	TEXT/58.S.B

List Price:	$144.00 $109.50
MAA Member Price:	$108.00 $82.13
AMS Member Price:	$108.00 $82.13

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Topology Through Inquiry

Michael Starbird : University of Texas at Austin, Austin, TX

Francis Su : Harvey Mudd College, Claremont, CA

MAA Press: An Imprint of the American Mathematical Society

Softcover ISBN:	978-1-4704-6261-1
Product Code:	TEXT/58.S

List Price:	$75.00
MAA Member Price:	$56.25
AMS Member Price:	$56.25

eBook ISBN:	978-1-4704-5532-3
Product Code:	TEXT/58.E

List Price:	$69.00
MAA Member Price:	$51.75
AMS Member Price:	$51.75

Softcover ISBN:	978-1-4704-6261-1
eBook ISBN:	978-1-4704-5532-3
Product Code:	TEXT/58.S.B

List Price:	$144.00 $109.50
MAA Member Price:	$108.00 $82.13
AMS Member Price:	$108.00 $82.13

Add to cart

Book Details
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AMS/MAA Textbooks

Volume: 58; 2019; 313 pp

MSC: Primary 54; 55

Topology Through Inquiry is a comprehensive introduction to point-set, algebraic, and geometric topology, designed to support inquiry-based learning (IBL) courses for upper-division undergraduate or beginning graduate students. The book presents an enormous amount of topology, allowing an instructor to choose which topics to treat. The point-set material contains many interesting topics well beyond the basic core, including continua and metrizability. Geometric and algebraic topology topics include the classification of 2-manifolds, the fundamental group, covering spaces, and homology (simplicial and singular). A unique feature of the introduction to homology is to convey a clear geometric motivation by starting with mod 2 coefficients.

The authors are acknowledged masters of IBL-style teaching. This book gives students joy-filled, manageable challenges that incrementally develop their knowledge and skills. The exposition includes insightful framing of fruitful points of view as well as advice on effective thinking and learning. The text presumes only a modest level of mathematical maturity to begin, but students who work their way through this text will grow from mathematics students into mathematicians.

Michael Starbird is a University of Texas Distinguished Teaching Professor of Mathematics. Among his works are two other co-authored books in the Mathematical Association of America's (MAA) Textbook series. Francis Su is the Benediktsson-Karwa Professor of Mathematics at Harvey Mudd College and a past president of the MAA. Both authors are award-winning teachers, including each having received the MAA's Haimo Award for distinguished teaching. Starbird and Su are, jointly and individually, on lifelong missions to make learning—of mathematics and beyond—joyful, effective, and available to everyone. This book invites topology students and teachers to join in the adventure.

Readership

Undergraduate and graduate students interested in topology and Inquiry Based Learning (IBL).

Cover
Title page
Copyright
Contents
Preface: Four Waysto Use This Book
(1) A textbook for introductory topology: * Potential road maps
(2) Topology courses beyond an introductory *course
(3) Independent study projects
(4) Joyful challenges for independent learners
A word about prerequisites
Acknowledgments
Introduction: The Enchanting World of Topology
Enticements to Topology
Learning to Create Mathematics
Introduction to Set-Theoretic Topology
Part 1 Point-Set Topology
Chapter 1. Cardinality: To Infinityand Beyond
1.1. Sets and Functions
1.2. Cardinality and Countable Sets
1.3. Uncountable Sets and Power Sets
1.4. The Schroeder-Bernstein Theorem
1.5. The Axiom of Choice
1.6. Ordinal Numbers
1.7. To Infinity and Beyond
Chapter 2. Topological Spaces: Fundamentals
2.1. Rubber Sheet Geometry and Special Sets
2.2. Open Sets and the Definition of a* Topological Space
2.3. Limit Points and Closed Sets
2.4. Interior and Boundary
2.5. Convergence of Sequences
2.6. Topological Essentials
Chapter 3. Bases, Subspaces, Products: Creating New Spaces
3.1. Bases
3.2. Subbases
3.3. Order Topology
3.4. Subspaces
3.5. Product Spaces
3.6. A Bounty of New Spaces
Chapter 4. Separation Properties: Separating This from That
4.1. Hausdorff, Regular, and Normal Spaces
4.2. Separation Properties and Products
4.3. A Question of Heredity
4.4. The Normality Lemma
4.5. Separating This from That
Chapter 5. Countable Features of Spaces: Size Restrictions
5.1. Separable Spaces, An Unfortunate Name
5.2. 2^{𝑛𝑑} Countable Spaces
5.3. 1^{𝑠𝑡} Countable Spaces
5.4. The Souslin Property
5.5. Count on It
Chapter 6. Compactness: The Next Best Thing to Being Finite
6.1. Compact Sets
6.2. The Heine-Borel Theorem
6.3. Compactness and Products
6.4. Countably Compact, Lindelöf Spaces
6.5. Paracompactness
6.6. Covering Up Reveals Strategies for * Producing Mathematics
Chapter 7. Continuity: When Nearby Points Stay Together
7.1. Continuous Functions
7.2. Properties Preserved by Continuous* Functions
7.3. Homeomorphisms
7.4. Product Spaces and Continuity
7.5. Quotient Maps and Quotient Spaces
7.6. Urysohn’s Lemma and the Tietze Extension Theorem
7.7. Continuity—Functions that Know Topology
Chapter 8. Connectedness: When Things Don’t Fall into Pieces
8.1. Connectedness
8.2. Cardinality, Separation Properties, and * Connectedness
8.3. Components and Continua
8.4. Path or Arcwise Connectedness
8.5. Local Connectedness
8.6. Totally Disconnected Spaces and the * Cantor Set
8.7. Hanging Together—Staying Connected
Chapter 9. Metric Spaces: Getting Some Distance
9.1. Metric Spaces
9.2. Continuous Functions between Metric* Spaces
9.3. Lebesgue Number Theorem
9.4. Complete Spaces
9.5. Metric Continua
9.6. Metrizability
9.7. Advanced Metrization Theorems
9.8. Paracompactness of Metric Spaces
9.9. Going the Distance
Part 2 Algebraic and Geometric Topology
Chapter 10. Transition From Point-Set Topology to Algebraic and Geometric Topology: Similar Strategies, Different Domains
10.1. Effective Thinking Principles—Strategies * for Creating Concepts
10.2. Onward: To Algebraic and Geometric* Topology
10.3. Manifolds and Complexes: Building* Locally, Studying Globally
10.4. The Homeomorphism Problem
10.5. Same Strategies, Different Flavors
Chapter 11. Classification of 2-Manifolds: Organizing Surfaces
11.1. Examples of 2-Manifolds
11.2. The Classification of 1-Manifolds
11.3. Triangulability of 2-Manifolds
11.4. The Classification of 2-Manifolds
11.5. The Connected Sum
11.6. Polygonal Presentations of 2-Manifolds
11.7. Another Classification of Compact* 2-Manifolds
11.8. Orientability
11.9. The Euler Characteristic
11.10. Manifolds with Boundary
11.11. Classifying 2-Manifolds: Going Below the Surface of Surfaces
Chapter 12. Fundamental Group:Capturing Holes
12.1. Invariants and Homotopy
12.2. Induced Homomorphisms and Invariance
12.3. Homotopy Equivalence and Retractions
12.4. Van Kampen’s Theorem
12.5. Lens Spaces
12.6. Knot Complements
12.7. Higher Homotopy Groups
12.8. The Fundamental Group—Not Such a * Loopy, Loopy Idea
Chapter 13. Covering Spaces:Layering It On
13.1. Basic Results and Examples
13.2. Lifts
13.3. Regular Covers and Cover Isomorphism
13.4. The Subgroup Correspondence
13.5. Theorems about Free Groups
13.6. Covering Spaces and 2-Manifolds
13.7. Covers are Cool
Chapter 14. Manifolds, Simplices, Complexes, and Triangulability: Building Blocks
14.1. Manifolds
14.2. Simplicial Complexes
14.3. Simplicial Maps and PL Homeomorphisms
14.4. Simplicial Approximation
14.5. Sperner’s Lemma and the Brouwer Fixed * Point Theorem
14.6. The Jordan Curve Theorem,* the Schoenflies Theorem,* and the Triangulability of 2-Manifolds
14.7. Simple Simplices; Complex Complexes; * Manifold Manifolds
Chapter 15. Simplicial ℤ₂-Homology: Physical Algebra
15.1. Motivation for Homology
15.2. Chains, Cycles, Boundaries, and the * Homology Groups
15.3. Induced Homomorphisms and Invariance
15.4. The Mayer-Vietoris Theorem
15.5. Introduction to Cellular Homology
15.6. Homology Is Easier Than It Seems
Chapter 16. Applications of ℤ₂-Homology: A Topological Superhero
16.1. The No Retraction Theorem
16.2. The Brouwer Fixed Point Theorem
16.3. The Borsuk-Ulam Theorem
16.4. The Ham Sandwich Theorem
16.5. Invariance of Domain
16.6. An Arc Does Not Separate the Plane
16.7. A Ball Does Not Separate Rⁿ
16.8. The Jordan-Brouwer Separation Theorem
16.9. Z ₂-Homology—A Topological Superhero
Chapter 17. Simplicial ℤ-Homology: Getting Oriented
17.1. Orientation and Z -Homology
17.2. Relative Simplicial Homology
17.3. Some Homological Algebra
17.4. Useful Exact Sequences
17.5. Homotopy Invariance and Cellular* Homology—Same as Z ₂
17.6. Homology and the Fundamental Group
17.7. The Degree of a Map
17.8. The Lefschetz Fixed Point Theorem
17.9. Z -Homology—A Step in Abstraction
Chapter 18. Singular Homology: Abstracting Objects to Maps
18.1. Eilenberg-Steenrod Axioms
18.2. Singular Homology
18.3. Topological Invariance and the Homotopy * Axiom
18.4. Relative Singular Homology
18.5. Excision
18.6. A Singular Abstraction
Chapter 19. The End: A Beginning—Reflections on Topology and Learning
Appendix A. Group Theory Background
A.1. Group Theory
Index
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