基本拓扑学及应用ELEMENTARY TOPOLOGY AND APPLICATIONS

The material in this book is organized in such a way that the reader gets to significant applications quickly, and the emphasis is on the geometric understanding and use of new concepts. The theme of the book is that topology is really the language of modern mathematics.

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