拓扑学 9.5分
读书笔记 Part I GENERAL TOPOLOGY
TOP J

Chapter 1 Set Theory and Logic 1 Fundamental Concepts 2 Functions 3 Relations 4 The Integers and the Real Numbers 5 Cartesian Products 6 Finite Sets 7 Countable and Uncountable Sets 8 The Principle of Recursive Definition 9 Infinite Sets and the Axiom of Choice 10 Well-Ordered Sets 11 The Maximum Principle Supplementary Exercises: Well-Ordering

Chapter 2 Topological Spaces and Continuous Functions 12 Topological Spaces 13 Basis for a Topology 14 The Order Topology 15 The Product Topology on X x Y 16 The Subspace Topology 17 Closed Sets and Limit Points 18 Continuous Functions 19 The Product Topology 20 The Metric Topology 21 The Metric Topology (continued) *22 The Quotient Topology *Supplementary Exercises: Topological Groups

Chapter 3 Connectedness and Compactness 23 Connected Spaces 24 Connected Subspaces of the Real Line *25 Components and Local Connectedness 26 Compact Spaces 27 Compact Subspaces of the Real Line 28 Limit Point Compactness 29 Local Compactness *Supplementary Exercises: Nets

Chapter 4 Countability and Separation Axioms 30 The Countability Axioms 31 The Separation Axioms 32 Normal Spaces 33 The Urysohn Lemma 34 The Urysohn Metrization Theorem *35 The Tietze Extension Theorem *36 Imbeddings of Manifolds *Supplementary Exercises: Review of the Basics

Chapter 5 The Tychonoff Theorem 37 The Tychonoff Theorem 38 The Stone-Cech Compactification

Chapter 6 Metrization Theorems and Paracompactness 39 Local Finiteness 40 The Nagata-Smirnov Metrization Theorem 41 Paracompactness 42 The Smirnov Metrization Theorem

Chapter 7 Complete Metric Spaces and Function Spaces 43 Complete Metric Spaces *44 A Space-Filling Curve 45 Compactness in Metric Spaces 46 Pointwise and Compact Convergence 47 Ascoli's Theorem

Chapter 8 Baire Spaces and Dimension Theory 48 Baire Spaces *49 A Nowhere-Differentiable Function 50 Introduction to Dimension Theory *Supplementary Exercises: Locally Euclidean Spaces

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