Elementary Topology and Applications

✍ Scribed by Carlos R. Borges

• Publisher: World Scientific
• Year: 2000
• Tongue: English
• Leaves: 215
• Category: Library

✦ Synopsis

Based on the theme that topology is really the universal language of modern mathematics, Borges (mathematics, U. of California-Davis) introduces it to students who have a good grasp of fundamentals of logic, set theory, elementary analysis, and group theory. He gets rapidly to applications. His goal is to prepare students for further study in mathematics. He does not include bibliographic references. Annotation copyrighted by Book News, Inc., Portland, OR

✦ Table of Contents

ELEMENTARY TOPOLOGY AND APPLICATIONS
Contents
CHAPTER 0. SETS AND NUMBERS
0.1 Rudiments of Logic
0.2 Fundamentals of Set Description
0.3 Set Inclusion and Equality
0.4 An Axiom System for Set Theory
0.5 Unions and Intersections
0.6 Set Difference
0.7 Integers and Induction
0.8 Simple Cartesian Products
0.9 Relations
0.10 Functions
0.11 Sequences
0.12 Indexing Sets
0.13 Important Formulas
0.14 Inverse Functions
0.15 More Important Formulas
0.16 Partitions
0.17 Equivalence Relations, Partitions and Functions
0.18 General Cartesian Products
0.19 The Sixth Axiom (Axiom of Choice)
0.20 Well-Orders and Zorn
0.21 Yet More Important Formulas
0.22 Cardinality
CHAPTER 1. METRIC AND TOPOLOGICAL SPACES
1.1 Metrics and Topologies
1.2 Time out for Notation
1.3 Metrics Generate Topologies
1.4 Continuous Functions
1.5 Subspaces
1.6 Comparable Topologies
CHAPTER 2. FROM OLD TO NEW SPACES
2.1 Product Spaces
2.2 Product Metrics and Topologies
2.3 Quotient Spaces
2.4 Applications (Möbius Band, Klien Bottle, Torus, Projevtive Plane, etc.)
CHAPTER 3. VERY SPECIAL SPACES
3.1 Compact Spaces
3.2 Compactification (One-Point Only)
3.3 Complete Metric Spaces (Baire-Catergory, Branch Contraction Theorem and Applications of Roots of y = h(x) to Systems of Liner Equations)
3.4 Connected and Arcwise Connected Spaces
CHAPTER 4. FUNCTION SPACES
4.1 Function Space Topologies
4.2 Completeness and Compactness (Ascolui-Arzela Theorm, Picard's theorm, Peano's Theorem)
4.3 Approximation (Bernstein's polynomials, Stone-Weierstrass Approximation)
4.4 Function-Space Functions
CHAPTER 5. TOPOLOGICAL GROUPS
5.1 Elementary Structures
5.2 Topological Isomorphism Theorems
5.3 Quotient Group Recognition
5.4 Morphism Groups (Topological and Transformation Groups)
CHAPTER 6. SPECIAL GROUPS
6.1 Preliminaries
6.2 Groups of Matrices
6.3 Groups of Isometries
6.4 Relativity and Lorentz Transformations
CHAPTER 7. NORMALITY AND PARACOMPACTNESS
7.1 Normal Spaces (Urysohn's Lemma)
7.2 Paracompact Spaces (Partitions of Unity with and Application to Embedding of Manifolds in Euclidean Spaces)
CHAPTER 8. THE FUNDAMENTAL GROUP
8.1 Description of II1, ( X, b)
8.2 Elementary Facts about II1 (X, b)
8.3 Simplicial Complexes
8.4 Barycentric Subdivision
8.5 The Simplicial Approximation
8.6 The Fundamental Group of Polytopes
8.7 Graphs and Tees
APPENDIX A. SOME INEQUALITIES
APPENDIX B. BINOMIAL EQUALITIES
LIST OF SYMBOLS
INDEX