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Introduction to Topology: Pure and Applied, CourseSmart eTextbook

By Colin Adams, Robert Franzosa

Published by Pearson

Published Date: Nov 5, 2007

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Description

For juniors, seniors, and graduate students of various majors, taking a first course in topology.

 

This book introduces topology as an important and fascinating mathematics discipline.   Students learn first the basics of point-set topology, which is enhanced by the real-world application of these concepts to science, economics, and engineering as well as other areas of mathematics. The second half of the book focuses on topics like knots, robotics, and graphs. The text is written in an accessible way for a range of undergraduates to understand the usefulness and importance of the application of topology to other fields.
CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.

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

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Introduction to Topology: Pure and Applied, CourseSmart eTextbook
Format: Safari Book

$78.99 | ISBN-13: 978-0-13-713503-5