Real Variables with Basic Metric Space Topology

Robert B. Ash

Publisher:

Dover Publications

Publication Date:

2009

Number of Pages:

213

Format:

Paperback

Price:

10.95

ISBN:

9780486472201

Category:

Textbook

MAA Review
Table of Contents

We do not plan to review this book.

INTRODUCTION
Basic Terminology
Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets
Distance and Convergence
Minicourse in Basic Logic
Limit Points and Closure
Review Problems for Chapter 1
SOME BASIC TOPOLOGICAL PROPERTIES OF Rp
Unions and Intersections of Open and Closed Sets
Compactness
Some Applications of Compactness
Least Upper Bounds and Completeness
Review Problems for Chapter 2
UPPER AND LOWER LIMITS OF SEQUENCES OF REAL NUMBERS
Generalization of the Limit Concept
Some Properties of Upper and Lower Limits
Convergence of Power Series
Review Problems for Chapter 3
CONTINUOUS FUNCTIONS
Continuity: Ideas, Basic Terminology, Properties
Continuity and Compactness
Types of Discontinuities
The Cantor Set
Review Problems for Chapter 4
DIFFERENTIATION
The Derivative and Its Basic Properties
Additional Properties of the Derivative; Some Applications of the Mean Value Theorem
Review Problems for Chapter 5
RIEMANN-STIELTJES INTEGRATION
Definition of the Integral
Properties of the Integral
Functions of Bounded Variation
Some Useful Integration Theorems
Review Problems for Chapter 6
UNIFORM CONVERGENCE AND APPLICATIONS
Pointwise and Uniform Convergence
Uniform Convergence and Limit Operations
The Weierstrass M-test and Applications
Equicontinuity and the Arzela-Ascoli Theorem
The Weierstrass Approximation Theorem
Review Problems for Chapter 7
FURTHER TOPOLOGICAL RESULTS
The Extension Problem
Baire Category Theorem
Connectedness
Semicontinuous Functions
Review Problems for Chapter 8
EPILOGUE
Some Compactness Results
Replacing Cantor's Nested Set Property
The Real Numbers Revisited
SOLUTIONS TO PROBLEMS
INDEX