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.
Preface I Euclidean Topology 1. Introduction to Topology 1.1 Deformations 1.2 Topological Spaces 2. Metric Topology in Euclidean Space 2.1 Distance 2.2 Continuity and Homeomorphism 2.3 Compactness and Limits 2.4 Connectedness 2.5 Metric Spaces in General 3. Vector Fields in the Plane 3.1 Trajectories and Phase Portraits 3.2 Index of a Critical Point 3.3 *Nullclines and Trapping Regions II Abstract Topology with Applications 4. Abstract Point-Set Topology 4.1 The Definition of a Topology 4.2 Continuity and Limits 4.3 Subspace Topology and Quotient Topology 4.4 Compactness and Connectedness 4.5 Product and Function Spaces 4.6 *The Infinitude of the Primes 5. Surfaces 5.1 Surfaces and Surfaces-with-Boundary 5.2 Plane Models and Words 5.3 Orientability 5.4 Euler Characteristic 6. Applications in Graphs and Knots 6.1 Graphs and Embeddings 6.2 Graphs, Maps, and Coloring Problems 6.3 Knots and Links 6.4 Knot Classification III Basic Algebraic Topology 7. The Fundamental Group 7.1 Algebra of Loops 7.2 Fundamental Group as Topological Invariant 7.3 Covering Spaces and the Circle 7.4 Compact Surfaces and Knot Complements 7.5 *Higher Homotopy Groups 8. Introduction to Homology 8.1 Rational Homology 8.2 Integral Homology Appendixes A. Review of Set Theory and Functions A.1 Sets and Operations on Sets A.2 Relations and Functions B. Group Theory and Linear Algebra B.1 Groups B.2 Linear Algebra C. Selected Solutions D. Notations Bibliography Index
""A 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 textbook's clear organization, worked examples, and concise writing style support a thorough understanding of basic topological principles, and might reasonably be expected to become a standard reference for teaching backgrounds of topology in the years to come.""
