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| 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 hnportant 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 Metrics andTopologies 2 Time out for Notation 3 Metrics Generate Topologies 4 Continuous Functions 5 Subspaces 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 (Mbius Band, Klein Bottle, Torus, Projective Plane, etc.) CHAPTER 3. VERY SPECIAL SPACES 3.1 Compact Spaces 3.2 Compactification (One-Point Only) 3.3 Complete Metric Spaces (Baire-Category, Banach Contraction Theorem and Applications of Roots of y = h(x) to Systems of Linear Equations 3.4 Connected and Arcwise Connected Spaces CHAPTER 4. FUNCTION SPACES 4.1 Function Space Topologies 4.2 Completness and Compactness (Ascoli-Arzela Theorem, Picard's Theorem, 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 Isometrics 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 APPENDLX A.SOME INEQUALITES APPENDIX B.BINOMIAL EQUALITIES LIST OF SYMBOLS INDEX |
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