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Book DetailsAMS/MAA TextbooksVolume: 58; 2019; 313 ppMSC: Primary 54; 55;
Topology Through Inquiry is a comprehensive introduction to pointset, algebraic, and geometric topology, designed to support inquirybased learning (IBL) courses for upperdivision undergraduate or beginning graduate students. The book presents an enormous amount of topology, allowing an instructor to choose which topics to treat. The pointset material contains many interesting topics well beyond the basic core, including continua and metrizability. Geometric and algebraic topology topics include the classification of 2manifolds, 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 IBLstyle teaching. This book gives students joyfilled, 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 coauthored books in the Mathematical Association of America's (MAA) Textbook series. Francis Su is the BenediktssonKarwa Professor of Mathematics at Harvey Mudd College and a past president of the MAA. Both authors are awardwinning 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.ReadershipUndergraduate and graduate students interested in topology and Inquiry Based Learning (IBL).

Table of Contents

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 SetTheoretic Topology

Part 1 PointSet 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 SchroederBernstein 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 HeineBorel 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 PointSet 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 2Manifolds: Organizing Surfaces

11.1. Examples of 2Manifolds

11.2. The Classification of 1Manifolds

11.3. Triangulability of 2Manifolds

11.4. The Classification of 2Manifolds

11.5. The Connected Sum

11.6. Polygonal Presentations of 2Manifolds

11.7. Another Classification of Compact* 2Manifolds

11.8. Orientability

11.9. The Euler Characteristic

11.10. Manifolds with Boundary

11.11. Classifying 2Manifolds: 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 2Manifolds

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 2Manifolds

14.7. Simple Simplices; Complex Complexes; * Manifold Manifolds

Chapter 15. Simplicial ℤ₂Homology: 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 MayerVietoris Theorem

15.5. Introduction to Cellular Homology

15.6. Homology Is Easier Than It Seems

Chapter 16. Applications of ℤ₂Homology: A Topological Superhero

16.1. The No Retraction Theorem

16.2. The Brouwer Fixed Point Theorem

16.3. The BorsukUlam 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 JordanBrouwer Separation Theorem

16.9. Z ₂Homology—A Topological Superhero

Chapter 17. Simplicial ℤHomology: 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. EilenbergSteenrod 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

Back Cover


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Topology Through Inquiry is a comprehensive introduction to pointset, algebraic, and geometric topology, designed to support inquirybased learning (IBL) courses for upperdivision undergraduate or beginning graduate students. The book presents an enormous amount of topology, allowing an instructor to choose which topics to treat. The pointset material contains many interesting topics well beyond the basic core, including continua and metrizability. Geometric and algebraic topology topics include the classification of 2manifolds, 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 IBLstyle teaching. This book gives students joyfilled, 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 coauthored books in the Mathematical Association of America's (MAA) Textbook series. Francis Su is the BenediktssonKarwa Professor of Mathematics at Harvey Mudd College and a past president of the MAA. Both authors are awardwinning 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.
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 SetTheoretic Topology

Part 1 PointSet 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 SchroederBernstein 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 HeineBorel 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 PointSet 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 2Manifolds: Organizing Surfaces

11.1. Examples of 2Manifolds

11.2. The Classification of 1Manifolds

11.3. Triangulability of 2Manifolds

11.4. The Classification of 2Manifolds

11.5. The Connected Sum

11.6. Polygonal Presentations of 2Manifolds

11.7. Another Classification of Compact* 2Manifolds

11.8. Orientability

11.9. The Euler Characteristic

11.10. Manifolds with Boundary

11.11. Classifying 2Manifolds: 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 2Manifolds

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 2Manifolds

14.7. Simple Simplices; Complex Complexes; * Manifold Manifolds

Chapter 15. Simplicial ℤ₂Homology: 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 MayerVietoris Theorem

15.5. Introduction to Cellular Homology

15.6. Homology Is Easier Than It Seems

Chapter 16. Applications of ℤ₂Homology: A Topological Superhero

16.1. The No Retraction Theorem

16.2. The Brouwer Fixed Point Theorem

16.3. The BorsukUlam 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 JordanBrouwer Separation Theorem

16.9. Z ₂Homology—A Topological Superhero

Chapter 17. Simplicial ℤHomology: 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. EilenbergSteenrod 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

Back Cover