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Introduction to Topology: Pure and Applied

Colin Adams and Robert Franzosa
Pearson Prentice Hall
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

This rather unusual book aims to introduce topology as a mathematical discipline and to demonstrate the value of topological ideas in various areas of science, engineering and mathematics. It is intended to serve as a text for a one or two semester undergraduate course for students who have completed an introductory course in abstract mathematics or for mathematically mature students who have taken the basic calculus sequence.

A reader of this text will learn the basics of point set topology in the first seven chapters, and then be introduced to a variety of topics including knots, manifolds, dynamical systems, fixed point theorems, and graph theory. The authors regard point set topology as the core of the book. For each of the other topics they offer a introductions designed to elicit student interest and facilitate further investigation. Applications are sprinkled liberally throughout; they are well-selected and germane to real problems.

The authors’ approach to point set topology has a good deal of pedagogical merit. Point set topology can be rather a dry subject, but a mixture of applications and good illustrations here keep it lively. For example, digital topology (useful in digital image processing) and phenotype spaces (of value to molecular biologists) are considered just after the concept of topological space is introduced. Geographic Information Systems are introduced after an initial discussion of open and closed sets, boundaries and limit points. Discussions of the applications here are not particularly deep, but they help motivate students and broaden their thinking.

The treatment of point set topology is neither as complete nor as extensive as in more typical textbooks. So, for example, infinite product spaces and their complications are not considered, the separation axioms get a much abbreviated treatment, and there’s not much of the usual zoo of pathological examples. This seems eminently sensible to me; students needing more specialized background will have a more than adequate basis to build upon.

The mixture of topics in the remaining chapters gives the instructor a lot of options. One could spend the better part of a term on homotopy and degree theory, followed by fixed point theorems and applications to economics and game theory. Alternatively, one could concentrate on topological graph theory and emphasize applications to chemistry. The final chapter on manifolds and cosmology addresses the fascinating question of the shape of the universe and what the cosmic microwave background might tell us about it.

The book has plenty of exercises, many of them standard. A notable feature is an excellent set of references to applications of topology in seventeen different areas.

A minor complaint I have is that the pages in the book are over-full. I expect this is a consequence of trying to keep down the page count and cost, but many pages look too crowded. It also might have been desirable to add a chapter on homology since there are many new and exciting applications there.

Overall, this is a creative and valuable addition to the world of introductory topology textbooks.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

0. Introduction

0.1. What is Topology and How is it Applied?
 0.2. A Glimpse at the History
 0.3. Sets and Operations on Them
 0.4. Euclidean Space
 0.5. Relations
 0.6. Functions

1. Topological Spaces
1.1. Open Sets and the Definition of a Topology
 1.2. Basis for a Topology
 1.3. Closed Sets
 1.4. Examples of Topologies in Applications

2. Interior, Closure, and Boundary
2.1. Interior and Closure of Sets
 2.2. Limit Points
 2.3. The Boundary of a Set
 2.4. An Application to Geographic Information Systems

3. Creating New Topological Spaces
3.1. The Subspace Topology
 3.2. The Product Topology
 3.3. The Quotient Topology
 3.4. More Examples of Quotient Spaces
 3.5. Configuration Spaces and Phase Spaces

4. Continuous Functions and Homeomorphisms
4.1. Continuity
 4.2. Homeomorphisms
 4.3. The Forward Kinematics Map in Robotics

5. Metric Spaces
5.1. Metrics
 5.2. Metrics and Information
 5.3. Properties of Metric Spaces
 5.4. Metrizability

6. Connectedness
6.1. A First Approach to Connectedness
 6.2. Distinguishing Topological Spaces via Connectedness
 6.3. The Intermediate Value Theorem
 6.4. Path Connectedness
 6.5. Automated Guided Vehicles

7. Compactness
7.1. Open Coverings and Compact Spaces
 7.2. Compactness in Metric Spaces
 7.3. The Extreme Value Theorem
 7.4. Limit Point Compactness
 7.5. One-Point Compactifications
8. Dynamical Systems and Chaos
8.1. Iterating Functions
 8.2. Stability
 8.3. Chaos
 8.4. A Simple Population Model with Complicated Dynamics
 8.5. Chaos Implies Sensitive Dependence on Initial Conditions

9. Homotopy and Degree Theory
9.1. Homotopy
 9.2. Circle Functions, Degree, and Retractions
 9.3. An Application to a Heartbeat Model
 9.4. The Fundamental Theorem of Algebra
 9.5. More on Distinguishing Topological Spaces
 9.6. More on Degree

10. Fixed Point Theorems and Applications
10.1. The Brouwer Fixed Point Theorem
 10.2. An Application to Economics
 10.3. Kakutani's Fixed Point Theorem
 10.4. Game Theory and the Nash Equilibrium

11. Embeddings
11.1. Some Embedding Results
 11.2. The Jordan Curve Theorem
 11.3. Digital Topology and Digital Image Processing

12. Knots
12.1. Isotopy and Knots
 12.2. Reidemeister Moves and Linking Number
 12.3. Polynomials of Knots
 12.4. Applications to Biochemistry and Chemistry
13. Graphs and Topology
13.1. Graphs
 13.2. Chemical Graph Theory
 13.3. Graph Embeddings
 13.4. Crossing Number and Thickness

14. Manifolds and Cosmology
14.1. Manifolds
 14.2. Euler Characteristic and the Classification of Compact Surfaces
 14.3. Three-Manifolds
 14.4. The Geometry of the Universe
 14.5. Determining which Manifold is the Universe

Additional Readings


Marin99's picture

Very well made, but lacks the solutions to the many exercices.