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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] 电子书 下载 mobi epub pdf txt

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编辑本段

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1 The General Topology of Dynamical Systems, Ethan Akin (1993, ISBN 978-0-8218-4932-3)[1]

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主要性质

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设X是非空集合，令J0={X，}，称（X，J0）为平庸拓扑空间，J0为平庸拓扑。令J1={A|AÌX}，称（X，J1）为离散拓扑空间。在离散拓扑空间中任意子集均是开集。对实数集R1，令J={BÌR1|"x∈G，∈ε>0，使（x－ε，x+ε）ÌG}，则（R1，J）就是一维欧几里得空间。类似地可定义n维欧几里得空间Rn。

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不动点问题

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分离公理展开

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②有限个开集的交是开集。

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拓扑空间（topological space），赋予拓扑结构的集合。如果对一个非空集合X给予适当的结构，使之能引入微积分中的极限和连续的概念，这样的结构就称为拓扑，具有拓扑结构的空间称为拓扑空间。引入拓扑结构的方法有多种，如邻域系、开集系、闭集系、闭包系、内部系等不同方法。

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东西不错，希望一直好用。

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