Understanding 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.""