**AMS/MAA Textbooks**

Volume: 58;
2019;
313 pp;
Softcover

MSC: Primary 54; 55;

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

List Price: $59.00

AMS Member Price: $44.25

MAA Member Price: $44.25

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

List Price: $59.00

AMS Member Price: $44.25

MAA Member Price: $44.25

#### Supplemental Materials

# Topology Through Inquiry

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*Michael Starbird; Francis Su*

MAA Press: An Imprint of the American Mathematical Society

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.

#### Table of Contents

# Table of Contents

## Topology Through Inquiry

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface: Four Waysto Use This Book ix11
- Introduction: The Enchanting World of Topology 117
- Part 1 Point-Set Topology 723
- Chapter 1. Cardinality: To Infinityand Beyond 925
- Chapter 2. Topological Spaces: Fundamentals 2743
- Chapter 3. Bases, Subspaces, Products: Creating New Spaces 4157
- Chapter 4. Separation Properties: Separating This from That 5571
- Chapter 5. Countable Features of Spaces: Size Restrictions 6581
- Chapter 6. Compactness: The Next Best Thing to Being Finite 7187
- Chapter 7. Continuity: When Nearby Points Stay Together 8197
- 7.1. Continuous Functions 8197
- 7.2. Properties Preserved by Continuous* Functions 84100
- 7.3. Homeomorphisms 86102
- 7.4. Product Spaces and Continuity 88104
- 7.5. Quotient Maps and Quotient Spaces 89105
- 7.6. Urysohn’s Lemma and the Tietze Extension Theorem 94110
- 7.7. Continuity—Functions that Know Topology 99115

- Chapter 8. Connectedness: When Things Don’t Fall into Pieces 101117
- 8.1. Connectedness 102118
- 8.2. Cardinality, Separation Properties, and * Connectedness 105121
- 8.3. Components and Continua 106122
- 8.4. Path or Arcwise Connectedness 111127
- 8.5. Local Connectedness 112128
- 8.6. Totally Disconnected Spaces and the * Cantor Set 115131
- 8.7. Hanging Together—Staying Connected 117133

- Chapter 9. Metric Spaces: Getting Some Distance 119135
- 9.1. Metric Spaces 119135
- 9.2. Continuous Functions between Metric* Spaces 123139
- 9.3. Lebesgue Number Theorem 124140
- 9.4. Complete Spaces 125141
- 9.5. Metric Continua 127143
- 9.6. Metrizability 128144
- 9.7. Advanced Metrization Theorems 129145
- 9.8. Paracompactness of Metric Spaces 133149
- 9.9. Going the Distance 133149

- Part 2 Algebraic and Geometric Topology 135151
- Chapter 10. Transition From Point-Set Topology to Algebraic and Geometric Topology: Similar Strategies, Different Domains 137153
- Chapter 11. Classification of 2-Manifolds: Organizing Surfaces 145161
- 11.1. Examples of 2-Manifolds 145161
- 11.2. The Classification of 1-Manifolds 149165
- 11.3. Triangulability of 2-Manifolds 149165
- 11.4. The Classification of 2-Manifolds 150166
- 11.5. The Connected Sum 157173
- 11.6. Polygonal Presentations of 2-Manifolds 158174
- 11.7. Another Classification of Compact* 2-Manifolds 159175
- 11.8. Orientability 162178
- 11.9. The Euler Characteristic 164180
- 11.10. Manifolds with Boundary 166182
- 11.11. Classifying 2-Manifolds: Going Below the Surface of Surfaces 168184

- Chapter 12. Fundamental Group:Capturing Holes 169185
- 12.1. Invariants and Homotopy 170186
- 12.2. Induced Homomorphisms and Invariance 180196
- 12.3. Homotopy Equivalence and Retractions 181197
- 12.4. Van Kampen’s Theorem 185201
- 12.5. Lens Spaces 189205
- 12.6. Knot Complements 192208
- 12.7. Higher Homotopy Groups 195211
- 12.8. The Fundamental Group—Not Such a * Loopy, Loopy Idea 196212

- Chapter 13. Covering Spaces:Layering It On 197213
- Chapter 14. Manifolds, Simplices, Complexes, and Triangulability: Building Blocks 207223
- 14.1. Manifolds 207223
- 14.2. Simplicial Complexes 210226
- 14.3. Simplicial Maps and PL Homeomorphisms 213229
- 14.4. Simplicial Approximation 215231
- 14.5. Sperner’s Lemma and the Brouwer Fixed * Point Theorem 219235
- 14.6. The Jordan Curve Theorem,* the Schoenflies Theorem,* and the Triangulability of 2-Manifolds 221237
- 14.7. Simple Simplices; Complex Complexes; * Manifold Manifolds 227243

- Chapter 15. Simplicial ℤ₂-Homology: Physical Algebra 229245
- Chapter 16. Applications of ℤ₂-Homology: A Topological Superhero 251267
- 16.1. The No Retraction Theorem 251267
- 16.2. The Brouwer Fixed Point Theorem 252268
- 16.3. The Borsuk-Ulam Theorem 252268
- 16.4. The Ham Sandwich Theorem 253269
- 16.5. Invariance of Domain 254270
- 16.6. An Arc Does Not Separate the Plane 254270
- 16.7. A Ball Does Not Separate Rⁿ 256272
- 16.8. The Jordan-Brouwer Separation Theorem 258274
- 16.9. Z ₂-Homology—A Topological Superhero 262278

- Chapter 17. Simplicial ℤ-Homology: Getting Oriented 263279
- 17.1. Orientation and Z -Homology 264280
- 17.2. Relative Simplicial Homology 270286
- 17.3. Some Homological Algebra 273289
- 17.4. Useful Exact Sequences 275291
- 17.5. Homotopy Invariance and Cellular* Homology—Same as Z ₂ 276292
- 17.6. Homology and the Fundamental Group 277293
- 17.7. The Degree of a Map 278294
- 17.8. The Lefschetz Fixed Point Theorem 279295
- 17.9. Z -Homology—A Step in Abstraction 281297

- Chapter 18. Singular Homology: Abstracting Objects to Maps 283299
- Chapter 19. The End: A Beginning—Reflections on Topology and Learning 295311
- Appendix A. Group Theory Background 299315
- Index 307323
- Back Cover Back Cover1330