Understanding topology : ǂa ǂpractical introduction

A fresh approach to topology makes this complex topic easier for students to master. 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. Life flourishes in the Book, a world of stacked realms spanning the ages, like the pages of an epic chronicle. Each page opens to domains evoking the glory of ancient empires, the passion of the Renaissance, or the adventure of court intrigue. Those who can travel between them are page turners, blessed with the power to go from one page to the next. For investigator Fiona Thorne, turning the page is normal life. Solving mysteries is where the excitement lives. No case is too small to ignite her curiosity, no page too familiar to explore. She is determined to leave her mark in the Book, no matter where her cases may lead. Whether unravelling conspiracies in an Elizabethan-like kingdom, outwitting sly cutthroat panthers on Roman seas, or navigating intrigues in a Venetian realm of canals and secretive fae, Fiona follows clues through page after page. Blaze, the page of fire, is wasting away. Fire elementals are being smuggled out in waves, but by whom? Fiona is on the job and nothing will hold her back, not even the overbearing Travel Guild. They may run the Book, but they don't rule her. With the fire page and its creatures on the brink of winking out, she has to race to unravel who the smugglers are and how to stop them. Is there more to this case than it seems? Can Fiona read between the lines and solve it in time? This is the first of a brilliant new historical fantasy series in a world where people are more than their ability, their title, or their page. Come for the world-building and mysteries. Stay for the delightful characters whom you'll root for time and time again. This series is definitely a page turner.

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.

Topology 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 chemical dynamics 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; [and] provides a brief review of set theory and functions and a summation of essential topics in group theory and linear algebra. 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-- PrefaceI Euclidean Topology1. Introduction to Topology1.1 Deformations1.2 Topological Spaces2. Metric Topology in Euclidean Space2.1 Distance2.2 Continuity and Homeomorphism2.3 Compactness and Limits2.4 Connectedness2.5 Metric Spaces in General3. Vector Fields in the Plane3.1 Trajectories and Phase Portraits3.2 Index of a Critical Point3.3 *Nullclines and Trapping RegionsII Abstract Topology with Applications4. Abstract Point-Set Topology4.1 The Definition of a Topology4.2 Continuity and Limits4.3 Subspace Topology and Quotient Topology4.4 Compactness and Connectedness4.5 Product and Function Spaces4.6 *The Infinitude of the Primes5. Surfaces5.1 Surfaces and Surfaces-with-Boundary5.2 Plane Models and Words5.3 Orientability5.4 Euler Characteristic6. Applications in Graphs and Knots6.1 Graphs and Embeddings6.2 Graphs, Maps, and Coloring Problems6.3 Knots and Links6.4 Knot ClassificationIII Basic Algebraic Topology7. The Fundamental Group7.1 Algebra of Loops7.2 Fundamental Group as Topological Invariant7.3 Covering Spaces and the Circle7.4 Compact Surfaces and Knot Complements7.5 *Higher Homotopy Groups8. Introduction to Homology8.1 Rational Homology8.2 Integral HomologyAppendixesA. Review of Set Theory and FunctionsA.1 Sets and Operations on SetsA.2 Relations and FunctionsB. Group Theory and Linear AlgebraB.1 GroupsB.2 Linear AlgebraC. Selected SolutionsD. NotationsBibliographyIndex Life flourishes in the Book, a realm of stacked worlds, like the pages of a novel. Each more unique than the last. Those who can travel through them are page turners. Blessed with the power to go from one page to the next. For investigator Fiona Thorne, being a turner is normal life. Solving mysteries is where the excitement lives. Minor cases are still work to be done, people to be helped, and shes determined to make an impact in the Book, no matter what. Blaze, the page of fire, is wasting away. Fire elementals are being smuggled out in waves, but by whom? Fiona is on the job and nothing will hold her back, not even the overbearing Travel Guild. They may run the Book, but they dont rule her. With the fire page and its creatures on the brink of winking out, she has to race to unravel who the smugglers are and how to stop them. Is there more to this case than it seems? Can Fiona read between the lines and solve it in time? "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; [and] provides a brief review of set theory and functions and a summation of essential topics in group theory and linear algebra. 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"--Back cover