Textbook in PDF format


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:
Preface
Introduction to Topology
Metric Topology in Euclidean Space
Vector Fields in the Plane
Abstract Point-Set Topology
Surfaces
Applications in Graphs and Knots
The Fundamental Group
Introduction to Homology
Set Theory and Functions
Groups
Linear Algebra
Answers
Bibliography
Index