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Full Description
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.
Contents
Introduction: The enchanting world of topology
Point-set topology: Cardinality: To infinity and beyond
Topological spaces: Fundamentals
Bases, subspaces, products: Creating new spaces
Separation properties: Separating this from that
Countable features of spaces: Size restrictions
Compactness: The next best thing to being finite
Continuity: When nearby points stay together
Connectedness: When things don't fall into pieces
Metric spaces: Getting some distance
Algebraic and geometric topology: Transition from point-set topology to algebraic and geometric topology: Similar strategies, different domains
Classification of 2-manifolds: Organizing surfaces
Fundamental group: Capturing holes
Covering spaces: Layering it on
Manifolds, simpleces, complexes, and triangulability: Building blocks
Simplicial $\mathbb{Z}_2$-homology: Physical algebra
Applications of $\mathbb{Z}_2$-homology: A topological superhero
Simplicial $\mathbb{Z}$-homology: Getting oriented
Singular homology: Abstracting objects to maps
The end: A beginning--reflections on topology and learning
Appendix: Group theory background
Index
