拓扑空间 [Topological Spaces: From Distance to Neighborhood] epub pdf mobi txt 电子书 下载 2024

拓扑空间 [Topological Spaces: From Distance to Neighborhood] epub pdf mobi txt 电子书 下载 2024

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拓扑空间 [Topological Spaces: From Distance to Neighborhood] epub pdf mobi txt 电子书 下载 2024

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内容简介

《拓扑空间》是一部本科生学习拓扑空间的基础教程。引导读者很好的学习拓扑中有关几何的东西什么是最重要的。《拓扑空间》的内容分为三大部分，线和面、矩阵空间和拓扑空间。书中将大量的数学词汇概念囊括其中，不要求读者对简单定理或者集合知识十分了解，从而减少读者理解上的难度。收敛定理的应用在帮助读者抓住重点的同时，逐渐接触并理解拓扑的概念，书中的知识点步步逼近，前九节重在为本科生讲述矩阵空间的知识，同时也包括了大量的材料，这些将成为研究生学习的教程。

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目录

Preface
PART Ⅰ THE LINE AND THE PLANE
Chapter 1 What Topology Is About
Topological Equivalence
Continuity and Convergence
A Few Conventions
Extra: Topological Diversions
Exercises
Chapter 2 Axioms for R
Extra: Axiom Systems
Exercises
Chapter 3 Convergent Sequences and Continuity
Subsequences
Uniform Continuity
The Plane
Extra: Bolzano （1781-1848）
Exercises
ChaPter 4 Curves in the Plane
Curves
Homeomorphic Sets
Brouwer's Theorem
Extra: L.E.J. Brouwer （1881-1966）

PART Ⅱ METRI SPACES
Chapter 5 Metrics
Extra: Camille Jordan （1838-1922）
Exercises
Chapter 6 Open and Closed Sets
Subsets of a Metric Space
Collections of Sets
Similar Metrics
Interior and Closure
The Empty Set
Extra: Cantor （1845-1918）
Exercises
Chapter 7 Completeness
Extra: Meager Sets and the Mazur Game
Exercises
Chapter 8 Uniform Convergence
Extra: Spaces of Continuous Functions
Exercises
Chapter 9 Sequential Compactness
Extra: The p-adic Numbers
Exercises
Chapter 10 Convergent Nets
Inadequacy of Sequences
Convergent Nets
-Extra: Knots
Exercises
Chapter 11 Transition to TOpology
Generalized Convergence
Topologies
Extra: The Emergence of the Professional Mathematician
Exercises

PART Ⅲ TOPOLOGICAL SPACES
Chapter 12 Topological Spaces
Extra: Map Coloring
Exercises
Chapter 13 Compactness and the Hausdorff Property
Compact Spaces
Hausdorff Spaces
Extra: Hausdorff and the Measure Problem
Exercises
Chapter 14 Products and Quotients
Product Spaces
Quotient Spaces
Extra: Surfaces
Exercises
Chapter 15 The Hahn-Tietze-Tong-Urysohn Theorems
Urysohn's Lemma
Interpolation and Extension
Extra: Nonstandard Mathematics
Exercises
Chapter 16 Connectedness
Connected Spaces
The Jordan Theorem
Extra: Continuous Deformation of Curves
Exercises
Chapter 17 Tvchonoffs Theorem
Extra: The Axiom of Choice
Exercises

PAler Ⅳ PosTsciuer
Chapter 18 A Smorgasbord for Further Study
Countability Conditions
Separation Conditions
Compactness Conditions
Compactifications
Connectivity Conditions
Extra: Dates from the History of General Topology
Exercises
Chapter 19 Countable Sets
Extra: The Continuum Hypothesis
A Farewell to the Reader
Literature
Index of Symbols
Index of Terms

前言/序言

拓扑空间 [Topological Spaces: From Distance to Neighborhood] epub pdf mobi txt 电子书 下载 2024

拓扑空间 [Topological Spaces: From Distance to Neighborhood] 下载 epub mobi pdf txt 电子书 2024

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东西不错，~~

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满足：

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目录

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目录

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③X、空集在J中，则称J是X的一个拓扑，J中的元称为开集，X连同拓扑J称为一个拓扑空间，记为（X，J）。

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主要性质

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拓扑学的需要大大刺激了抽象代数学的发展，并且形成了两个新的代数学分支：同调代数与代数K理论。代数几何学从50年代以来已经完全改观。托姆的配边理论直接促使代数簇的黎曼－罗赫定理的产生，后者又促使拓扑K 理论的产生。现代代数几何学已完全使用上同调的语言，代数数论与代数群也在此基础上取得许多重大成果，例如有关不定方程整数解数目估计的韦伊猜想和莫德尔猜想的证明。范畴与函子的观念，是在概括代数拓扑的方法论时形成的。范畴论已深入数学基础、代数几何学等分支，对拓扑学本身也有影响。如拓扑斯的观念大大拓广了经典的拓扑空间观念。

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于是度量空间都是拓扑空间。但不是所有拓扑空间都可定义度量，使得该度量下的开集族与原拓扑空间的开集族一致；详见度量化定理。

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