Understanding Topology: A Practical Introduction
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Topology—the branch of mathematics that studies the properties of spaces that remain unaffected by stretching and other distortions—can present significant challenges for undergraduate students of mathematics and the sciences. Understanding Topology aims to change that. The perfect introductory topology textbook, Understanding Topology requires only a knowledge of calculus and a general familiarity with set theory and logic. Equally approachable and rigorous, the book's clear organization, worked examples, and concise writing style support a thorough understanding of basic topological principles. Professor Shaun V. Ault's unique emphasis on fascinating applications, from mapping DNA to determining the shape of the universe, will engage students in a way traditional topology textbooks do not. This groundbreaking new text: • presents Euclidean, abstract, and basic algebraic topology • explains metric topology, vector spaces and dynamics, point-set topology, surfaces, knot theory, graphs and map coloring, the fundamental group, and homology • includes worked example problems, solutions, and optional advanced sections for independent projects Following a path that will work with any standard syllabus, the book is arranged to help students reach that "Aha!" moment, encouraging readers to use their intuition through local-to-global analysis and emphasizing topological invariants to lay the groundwork for algebraic topology. Contents......Page 3 Preface......Page 5 01 Introduction to Topology......Page 10 02 Metric Topology in Euclidean Space......Page 35 03 Vector Fields in the Plane......Page 94 04 Abstract Point-Set Topology......Page 125 05 Surfaces......Page 173 06 Applications in Graphs and Knots......Page 213 07 The Fundamental Group......Page 262 08 Introduction to Homology......Page 304 Appendix A - Set Theory and Functions......Page 326 Appendix B1 - Groups......Page 356 Appendix B2 - Linear Algebra......Page 361 Answers......Page 373 Bibliography......Page 396 Index......Page 400
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