J.史迪威著的《實數（英文版）》是Undergraduate Texts in Mathematics系列叢書之一，與多數簡述實數的教材不同，本書則論述了實數的方方面面，特別聚焦分析集合論，把闡述分析的精髓和介紹集合論完美的結合在在一起，書中還涉及數學史內容。 本書主要面向掌握微積分等基本數學知識的大學高年級本科生，也適用於研究生和數學工作者。
作者介紹
(美)J.史迪威
目錄
Preface 1 The Fundamental Questions 1.1 A Specific Question: Why Does ab = ba? 1.2 What Are Numbers? 1.3 What Is the Line? 1.4 What Is Geometry? 1.5 What Are Functions? 1.6 What Is Continuity? 1.7 What Is Measure? 1.7.1 Area and Volume 1.8 What Does Analysis Want from R? 1.9 Historical Remarks 2 From Discrete to Continuous 2.1 Counting and Induction 2.2 Induction and Arithmetic 2.2.1 Addition 2.2.2 Multiplication 2.2.3 The Law ab = ba Revisited 2.3 From Rational to Real Numbers 2.3.1 Visualizing Dedekind Cuts 2.4 Arithmetic of Real Numbers 2.4.1 The Square Root of 2 2.4.2 The Equation X/2X/3 = vr6 2.5 Order and Algebraic Properties 2.5.1 Algebraic Properties of g 2.6 Other Completeness Properties 2.7 Continued Fractions 2.8 Convergence of Continued Fractions 2.9 Historical Remarks 2.9.1 R as a Complete Ordered Field 3 Infinite Sets 3.1 Countably Infinite Sets 3.1.1 The Universal Library 3.2 An Explicit Bijection Between lq and N2 3.3 Sets Equinumerous with R 3.4 The Cantor-Schrtder-Bernstein Theorem 3.4.1 More Sets Equinumerous with R 3.4.2 The Universal Jukebox 3.5 The Uncountability of R 3.5.1 The Diagonal Argument 3.5.2 The Measure Argument 3.6 Two Classical Theorems About Infinite Sets 3.7 The Cantor Set 3.7.1 Measure of the Cantor Set 3.8 Higher Cardinalities 3.8.1 The Continuum Hypothesis 3.8.2 Extremely High Cardinalities 3.9 Historical Remarks 4 Functions and Limits 4.1 Convergence of Sequences and Series
4.1.1 Divergent and Conditionally Convergent Series 4.2 Limits and Continuity 4.3 Two Properties of Continuous Functions 4.3.1 The Devil's Staircase 4.4 Curves 4.4.1 - A Curve Without Tangents 4.4.2 A Space-Filling Curve 4.5 Homeomorphisms 4.6 Uniform Convergence 4.7 Uniform Continuity 4.8 The Riemann Integral 4.8.1 The Fundamental Theorem of Calculus 4.9 Historical Remarks 5 Open Sets and Continuity 5.1 Open Sets 5.2 Continuity via Open Sets 5.2.1 The General Concept of Open Set 5.3 Closed Sets 5.4 Compact Sets 5.5 Perfect Sets 5.5.1 Beyond Open and Closed Sets 5.6 Open Subsets of the Irrationals 5.6.1 Encoding Open Subsets of At" by Elements of N 5.7 Historical Remarks 6 Ordinals 6.1 Counting Past Infinity 6.2 What Are Ordinals? 6.2.1 Finite Ordinals 6.2.2 Infinite Ordinals: Successor and Least Upper Bound 6.2.3 Uncountable ordinals 6.3 Well-ordering and Transfinite Induction 6.4 The Cantor-Bendixson Theorem 6.5 The ZF Axioms for Set Theory 6.6 Finite Set Theory and Arithmetic 6.7 The Rank Hierarchy 6.7.1 Cardinality 6.8 Large Sets 6.9 Historical Remarks 7 The Axiom of Choice 7.1 Some Naive Questions About Infinity 7.2 The Full Axiom of Choice and Well-Ordering 7.2.1 Cardinal Numbers 7.3 The Continuum Hypothesis 7.4 Filters and Ultrafilters 7.5 Games and Winning Strategies 7.6 Infinite Games 7.6.1 Strategies 7.7 The Countable Axiom of Choice 7.8 Zorn's Lemma 7.9 Historical Remarks
7.9.1 AC, AD, and the Natural Numbers 8 Borel Sets 8.1 Borel Sets 8.2 Borel Sets and Continuous Functions 8.3 Universal ]~a Sets 8.4 The Borel Hierarchy 8.5 Bake Functions 8.6 The Number of Borel Sets 8.7 Historical Remarks 9 Measure Theory 9.1 Measure of Open Sets 9.2 Approximation and Measure 9.3 Lebesgue Measure 9.4 Functions Continuous Almost Everywhere 9.4.1 Uniform a-Continuity 9.5 Riemann Integrable Functions 9.6 Vitali's Nonmeasurable Set 9.7 Ultrafilters and Nonmeasurable Sets 9.8 Historical Remarks 10 Reflections 10.1 What Are Numbers? 10.2 What Is the Line? 10.3 What Is Geometry? 10.4 What Are Functions? 10.5 What Is Continuity? 10.6 What Is Measure? 10.7 What Does Analysis Want from R? 10.8 Further Reading 10.8.1 Greek Mathematics 10.8.2 The Number Concept 10.8.3 Analysis 10.8.4 Set Theory 10.8.5 Axiom of Choice References Index
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