拓扑学

本书作者在拓扑学领域享有盛誉。 本书分为两个独立的部分；第一部分普通拓扑学，讲述点集拓扑学的内容；前4章作为拓扑学的引论，介绍作为核心题材的集合论、拓扑空间。连通性、紧性以及可数性和分离性公理；后4章是补充题材；第二部分代数拓扑学，讲述与拓扑学核心题材相关的主题，其中包括基本群和覆盖空间及其应用。 本书最大的特点在于对理论的清晰阐述和严谨证明，力求让读者能够充分理解。对于疑难的推理证明，将其分解为简化的步骤，不给读者留下疑惑。此外，书中还提供了大量练习，可以巩固加深学习的效果。严格的论证，清晰的条理、丰富的实例，让深奥的拓扑学变得轻松易学。

Preface A Note to the Reader Part I GENERAL TOPOLOGY 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 Part II ALGEBRAIC TOPOLOGY Chapter 9 The Fundamental Group 51 Homotopy of Paths 52 The Fundamental Group 53 Covering Spaces 54 The Fundamental Group of the Circle 55 Retractions and Fixed Points *56 The Fundamental Theorem of Algebra *57 The Borsuk-Ulam Theorem 58 Deformation Retracts and Homotopy Type 59 The Fundamental Group of Sn 60 Fundamental Groups of Some Surfaces Chapter 10 Separation Theorems in the Plane 61 The Jordan Separation Theorem *62 Invariance of Domain 63 The Jordan Curve Theorem 64 Imbedding Graphs in the Plane 65 The Winding Number of a Simple Closed Curve 66 The Cauchy Integral Formula Chapter 11 The Seifert-van Kampen Theorem 67 Direct Sums of Abelian Groups 68 Free Products of Groups 69 Free Groups 70 The Seifert-van Kampen Theorem 71 The Fundamental Group of a Wedge of Circles 72 Adjoining a Two-cell 73 The Fundamental Groups of the Torus and the Dunce Cap Chapter 12 Classification of Surfaces 74 Fundamental Groups of Surfaces 75 Homology of Surfaces 76 Cutting and Pasting 77 The Classification Theorem 78 Constructing Compact Surfaces Chapter 13 Classification of Covering Spaces 79 Equivalence of Covering Spaces 80 The Universal Covering Space *81 Covering Transformations 82 Existence of Covering Spaces *Supplementary Exercises: Topological Properties and Chapter 14 Applications to Group Theory 83 Covering Spaces of a Graph 84 The Fundamental Group of a Graph 85 Subgroups of Free Groups Bibliography Index

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• 很好

  来源：豆瓣 发布时间：2025-05-11 21:46:32

• 拓扑入门好书

  来源：豆瓣 发布时间：2025-05-11 21:46:32