Introduction to Topology
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For juniors and seniors 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. Product Description
For juniors and seniors 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. Features + Benefits
Theoretical and applied approach- the authors focus on the basic concepts of topology and their utility using real-world applications
--Applications cover a wide variety of disciplines, including molecular biology, digital image processing, robotics, population dynamics, medicine, economics, synthetic chemistry, electronic circuit design, and cosmology.
Intuitive and accessibly written text
--Rigorous presentation of the mathematics with intuitive descriptions and discussions to increase student understand.
--Examples of real world application keep students engrossed in the material
Numerous figures allow students to visualize and understand the material presented
For more information about the text, the authors, and for errata, please visit:
http://germain.its.maine.edu/~franzosa/ITPA.htm
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
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. Product Description
For juniors and seniors 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. Features + Benefits
Theoretical and applied approach- the authors focus on the basic concepts of topology and their utility using real-world applications
--Applications cover a wide variety of disciplines, including molecular biology, digital image processing, robotics, population dynamics, medicine, economics, synthetic chemistry, electronic circuit design, and cosmology.
Intuitive and accessibly written text
--Rigorous presentation of the mathematics with intuitive descriptions and discussions to increase student understand.
--Examples of real world application keep students engrossed in the material
Numerous figures allow students to visualize and understand the material presented
For more information about the text, the authors, and for errata, please visit:
http://germain.its.maine.edu/~franzosa/ITPA.htm
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
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.
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