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內容簡介

《拓撲空間》是一部本科生學習拓撲空間的基礎教程。引導讀者很好的學習拓撲中有關幾何的東西什麼是最重要的。《拓撲空間》的內容分為三大部分，綫和麵、矩陣空間和拓撲空間。書中將大量的數學詞匯概念囊括其中，不要求讀者對簡單定理或者集閤知識十分瞭解，從而減少讀者理解上的難度。收斂定理的應用在幫助讀者抓住重點的同時，逐漸接觸並理解拓撲的概念，書中的知識點步步逼近，前九節重在為本科生講述矩陣空間的知識，同時也包括瞭大量的材料，這些將成為研究生學習的教程。

內頁插圖

目錄

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

拓撲空間 [Topological Spaces: From Distance to Neighborhood] 下載 mobi epub pdf txt 電子書

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空間中一條自身不相交的封閉麯綫，會發生打結現象。要問一個結能否解開（即能否變形成平放的圓圈），或者問兩個結能否互變，並且不隻做個模型試試，還要給齣證明，那就遠不是件容易的事瞭（見紐結理論）。

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編輯本段

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注意到如能在X中給齣度量則自然在X中給齣拓撲（由度量決定的開集）。

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不動點問題

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除去七橋問題，四色問題，歐拉定理等，拓撲學中還有很多有趣並且很基本的問題。

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拓撲空間

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4 The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Russell A. Gordon (1994, ISBN 978-0-8218-3805-1)

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分類介紹

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分離公理展開

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