Introduction to Topology: Pure And Applied by Colin Adams

Introduction to Topology: Pure And Applied

byColin Adams, Robert Franzosa

Hardcover | June 18, 2007

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 Learn the basics of point-set topology with the understanding of its real-world application to a variety of other subjects including science, economics, engineering, and other areas of mathematics.   Introduces topology as an important and fascinating mathematics discipline to retain the readers interest in the subject. Is written in an accessible way for readers to understand the usefulness and importance of the application of topology to other fields. Introduces topology concepts combined with their real-world application to subjects such DNA, heart stimulation, population modeling, cosmology, and computer graphics. Covers topics including knot theory, degree theory, dynamical systems and chaos, graph theory, metric spaces, connectedness, and compactness. A useful reference for readers wanting an intuitive introduction to topology.

About The Author

Colin Adams is the Thomas T. Read Professor of Mathematics at Williams College. He received his PhD from the University of Wisconsin–Madison in 1983. He is particularly interested in the mathematical theory of knots, their applications, and their connections with hyperbolic geometry. He is the author of The Knot Book, an elementary in...
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Title:Introduction to Topology: Pure And AppliedFormat:HardcoverDimensions:512 pages, 9.3 × 7.1 × 1.3 inPublished:June 18, 2007Publisher:Pearson EducationLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0131848690

ISBN - 13:9780131848696

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Extra Content

Table of Contents

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 The One-Point Compactification


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