Principles of Topology (Dover Books on Mathematics) 1st edition by Fred Croom- Ebook PDF Instant Download/Delivery: 0486801543, 978-0486801544
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ISBN 10: 0486801543
ISBN 13: 978-0486801544
Author: Fred Croom
Topology is a natural, geometric, and intuitively appealing branch of mathematics that can be understood and appreciated by students as they begin their study of advanced mathematical topics. Designed for a one-semester introduction to topology at the undergraduate and beginning graduate levels, this text is accessible to students familiar with multivariable calculus. Rigorous but not abstract, the treatment emphasizes the geometric nature of the subject and the applications of topological ideas to geometry and mathematical analysis.
Customary topics of point-set topology include metric spaces, general topological spaces, continuity, topological equivalence, basis, subbasis, connectedness, compactness, separation properties, metrization, subspaces, product spaces, and quotient spaces. In addition, the text introduces geometric, differential, and algebraic topology. Each chapter includes historical notes to put important developments into their historical framework. Exercises of varying degrees of difficulty form an essential part of the text.
Principles of Topology (Dover Books on Mathematics) 1st Table of contents:
Chapter 1 Introduction
1.1 The Nature of Topology
1.2 The Origin of Topology
1.3 Preliminary Ideas from Set Theory
1.4 Operations on Sets: Union, Intersection, and Difference
1.5 Cartesian Products
1.6 Functions
1.7 Equivalence Relations
Chapter 2 The Line and the Plane
2.1 Upper and Lower Bounds
2.2 Finite and Infinite Sets
2.3 Open Sets and Closed Sets on the Real Line
2.4 The Nested Intervals Theorem
2.5 The Plane
Suggestions for Further Reading
Historical Notes for Chapter 2
Chapter 3 Metric Spaces
3.1 The Definition and Some Examples
3.2 Open Sets and Closed Sets in Metric Spaces
3.3 Interior, Closure, and Boundary
3.4 Continuous Functions
3.5 Equivalence of Metric Spaces
3.6 New Spaces from Old
3.7 Complete Metric Spaces
Suggestions for Further Reading
Historical Notes for Chapter 3
Chapter 4 Topological Spaces 99
4.1 The Definition and Some Examples
4.2 Interior; Closure, and Boundary
4.3 Basis and Subbasis
4.4 Continuity and Topological Equivalence
4.5 Subspaces
Suggestions for Further Reading
Historical Notes for Chapter 4
Chapter 5 Connectedness
5.1 Connected and Disconnected Spaces
5.2 Theorems on Connectedness
5.3 Connected Subsets of the Real Line
5.4 Applications of Connectedness
5.5 Path Connected Spaces
5.6 Locally Connected and Locally Path Connected Spaces Suggestions for Further Reading
Suggestions for Further Reading
Historical Notes for Chapter 5
Chapter 6 Compactness
6.1 Compact Spaces and Subspaces
6.2 Compactness and Continuity
6.3 Properties Related to Compactness
6.4 One-Point Compactification
6.5 The Cantor Set
Suggestions for Further Reading
Historical Notes for Chapter 6
Chapter 7 Product and Quotient Spaces
7.1 Finite Products
7.2 Arbitrary Products
7.3 Comparison of Topologies
7.4 Quotient Spaces
7.5 Surfaces and Manifolds
Suggestions for Further Reading
Historical Notes for Chapter 7
Chapter 8 Separation Properties and Metrization
8.1 T0, T1, and T2-Spaces
8.2 Regular Spaces
8.3 Normal Spaces
8.4 Separation by Continuous Functions
8.5 Metrization
8.6 The Stone-Cech Compactification
Suggestions for Further Reading
Historical Notes for Chapter 8
Chapter 9 The Fundamental Group
9.1 The Nature of Algebraic Topology
9.2 The Fundamental Group
9.3 The Fundamental Group of S1
9.4 Additional Examples of Fundamental Groups
9.5 The Brouwer Fixed Point Theorem and Related Results
9.6 Categories and Functors
Suggestions for Further Reading
Historical Notes for Chapter 9
Appendix: Introduction to Groups
Bibliography
Index
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