While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.
Author(s): John Stillwell
Series: Undergraduate Texts in Mathematics
Publisher: Springer Science & Business Media
Year: 2013
Language: English
Pages: 253
Preface
Acknowledgments
Contents
Chapter
1 The Fundamental Questions
1.1 A Specific Question: Why Does ab=ba?
Exercises
1.2 What Are Numbers?
Exercises
1.3 What Is the Line?
Exercises
1.4 What Is Geometry?
Exercises
1.5 What Are Functions?
Exercises
1.6 What Is Continuity?
Exercises
1.7 What Is Measure?
1.7.1 Area and Volume
Exercises
1.8 What Does Analysis Want from R?
Exercises
1.9 Historical Remarks
Chapter
2 From Discrete to Continuous
2.1 Counting and Induction
Exercises
2.2 Induction and Arithmetic
2.2.1 Addition
2.2.2 Multiplication
2.2.3 The Law ab=ba Revisited
Exercises
2.3 From Rational to Real Numbers
2.3.1 Visualizing Dedekind Cuts
Exercises
2.4 Arithmetic of Real Numbers
2.4.1 The Square Root of 2
2.4.2 The Equation 23=6
Exercises
2.5 Order and Algebraic Properties
2.5.1 Algebraic Properties of R
Exercises
2.6 Other Completeness Properties
Exercises
2.7 Continued Fractions
Exercises
2.8 Convergence of Continued Fractions
Exercises
2.9 Historical Remarks
2.9.1 R as a Complete Ordered Field
Chapter
3 Infinite Sets
3.1 Countably Infinite Sets
3.1.1 The Universal Library
Exercises
3.2 An Explicit Bijection Between N and N2
Exercises
3.3 Sets Equinumerous with R
Exercises
3.4 The Cantor–Schröder–Bernstein Theorem
3.4.1 More Sets Equinumerous with R
3.4.2 The Universal Jukebox
Exercises
3.5 The Uncountability of R
3.5.1 The Diagonal Argument
3.5.2 The Measure Argument
Exercises
3.6 Two Classical Theorems About Infinite Sets
Exercises
3.7 The Cantor Set
3.7.1 Measure of the Cantor Set
Exercises
3.8 Higher Cardinalities
3.8.1 The Continuum Hypothesis
3.8.2 Extremely High Cardinalities
Exercises
3.9 Historical Remarks
Chapter
4 Functions and Limits
4.1 Convergence of Sequences and Series
4.1.1 Divergent and Conditionally Convergent Series
Exercises
4.2 Limits and Continuity
Exercises
4.3 Two Properties of Continuous Functions
4.3.1 The Devil's Staircase
Exercises
4.4 Curves
4.4.1 A Curve Without Tangents
4.4.2 A Space-Filling Curve
Exercises
4.5 Homeomorphisms
Exercises
4.6 Uniform Convergence
Exercises
4.7 Uniform Continuity
Exercises
4.8 The Riemann Integral
4.8.1 The Fundamental Theorem of Calculus
Exercises
4.9 Historical Remarks
Chapter
5 Open Sets and Continuity
5.1 Open Sets
Exercises
5.2 Continuity via Open Sets
5.2.1 The General Concept of Open Set
Exercises
5.3 Closed Sets
Exercises
5.4 Compact Sets
Exercises
5.5 Perfect Sets
5.5.1 Beyond Open and Closed Sets
Exercises
5.6 Open Subsets of the Irrationals
5.6.1 Encoding Open Subsets of N by Elements of N
Exercises
5.7 Historical Remarks
Chapter
6 Ordinals
6.1 Counting Past Infinity
Exercises
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
Exercises
6.3 Well-Ordering and Transfinite Induction
Exercises
6.4 The Cantor–Bendixson Theorem
Exercises
6.5 The ZF Axioms for Set Theory
Exercises
6.6 Finite Set Theory and Arithmetic
Exercises
6.7 The Rank Hierarchy
6.7.1 Cardinality
Exercises
6.8 Large Sets
6.9 Historical Remarks
Chapter
7 The Axiom of Choice
7.1 Some Naive Questions About Infinity
Exercises
7.2 The Full Axiom of Choice and Well-Ordering
7.2.1 Cardinal Numbers
Exercises
7.3 The Continuum Hypothesis
Exercises
7.4 Filters and Ultrafilters
Exercises
7.5 Games and Winning Strategies
Exercises
7.6 Infinite Games
7.6.1 Strategies
Exercises
7.7 The Countable Axiom of Choice
Exercises
7.8 Zorn's Lemma
Exercises
7.9 Historical Remarks
7.9.1 AC, AD, and the Natural Numbers
Chapter 8 Borel Sets
8.1 Borel Sets
Exercises
8.2 Borel Sets and Continuous Functions
Exercises
8.3 Universal bold0mu mumu dottedα Sets
Exercises
8.4 The Borel Hierarchy
Exercises
8.5 Baire Functions
Exercises
8.6 The Number of Borel Sets
Exercises
8.7 Historical Remarks
Chapter
9 Measure Theory
9.1 Measure of Open Sets
Exercises
9.2 Approximation and Measure
Exercises
9.3 Lebesgue Measure
Exercises
9.4 Functions Continuous Almost Everywhere
9.4.1 Uniform α-Continuity
Exercises
9.5 Riemann Integrable Functions
Exercises
9.6 Vitali's Nonmeasurable Set
Exercises
9.7 Ultrafilters and Nonmeasurable Sets
Exercises
9.8 Historical Remarks
Chapter
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
