University of Rhode Island      Department of Mathematics

MTH 435  Mathematical Analysis and Topology I

Monday, Wednesday 3 - 4:15pm Lippitt Hall 204,
Friday 3-3:50pm Lippitt Hall 204



Instructor  Dr. Mark Comerford
Office  Lippitt 102 F
Phone  874 5984
Email  mcomerford@math.uri.edu
Office Hours
 Tuesday 2 - 4 pm
  or by appointment
Texts  Russell A. Gordon
Real Analysis - A First Course
(Second Edition),
 ISBN-10 0201437279

Theodore W. Gamelin, Robert Everist Greene
Introduction to Topology (Second Edition),
 ISBN 0-486-40680-6

Bert Mendelson
Introduction to Topology
(Third Edition),
 ISBN 0-486-66352-3

Prerequisites  Mth 243 or equivalent



Lectures and Homework Problems


Section Problems
0.1 Mathematical Logic Homework 1
0.2 Sets and Functions
0.3 Mathematical Induction
1.1 Real Numbers 10, 11, 14
I   Metric Spaces 2.2: 4
1.2 Absolute Value, Intervals and Inequalities 8, 17, 21
I  Open and Closed Sets, Neighbourhoods 2.4: 6
1.3 The Completeness Axiom 6, 19, 21
II   Density and Sequences in Metric Spaces TBA
1.4 Countable and Uncountable Sets 9, 5, 8
1.5 Real-Valued Functions
2.1 Convergent Sequences 5, 9
III Cauchy Sequences II, III Problems
2.2 Monotone and Cauchy Sequences 2, 27 (27 is extra credit)
2.3 Subsequences 12
IV   Limits, Continuity TBA
2.4 Supplementary Exercises 18
3.1 The Limit of a Function 10, 27
3.2 Continuous Functions 10
3.3 Intermediate and Extreme Values 7, 23
3.4 Uniform Continuity 4, 5
3.5 Monotone Functions
3.6 Supplementary Exercises
IV Completeness,
    the Contraction Mapping Principle
Completeness Problems
4.1 The Derivative of a Function 3
4.2 The Mean Value Theorem 10
4.3 Further Topics on Differentiation
4.4 Supplementary Exercises

 

Here is a link to Theorems 4.30 and 4.31 at the end of the course whose proofs we did not have time to cover in class.

A list of the theorems whose proofs you need to know for the final can be found here.

Exams


Midterm  3 - 4:15pm, Wednesday October 16  Lippitt 204
Final   3 - 6pm, Friday December 13   Lippitt 204

 

Evaluation


Homework   40%
Midterm   25%
Final   35%


Final Grade Calculation

A 95 - 100, A- 90 - 95, B+ 87 - 90, B 83 - 87, B- 80 - 83, C+ 77 - 80, C 73 - 77, C- 70 - 73, D+ 67 - 70, D 60 - 67, F < 60.



Course Description

This course is an introduction to the wonderful world of real numbers, real analysis and topology. The course is self-contained and so no initial knowledge of real analysis is required. However, familiarity with calculus concepts up to and including math 243 (multivariable calculus) is essential and math 307 (Introduction to Mathematical Rigor) is highly desirable. In real analysis we will cover the material in the first four chapters of Gordon's book up to differentiation and the mean value theorem while in topology we will cover the basic results on metric spaces.

In addition to covering the course material, there will be strong emphasis on mathematical rigor and mastering techniques of proof. This is reflected in the rather large percentage of the overall grade which is given over to homework which will be assigned weekly.

In addition to homework, there will be one midterm (provisional date Monday October 15) and a final. The material covered in the exams will be partly based on the homework, but there is also some expectation that there will be unseen problems. In addition, you may be asked to give proofs of theorems covered in class and you will be given in advance a list of those theorems which are examinable.


Goals and Objectives

The goals of the course are to have you develop the skills of working with real numbers, limits, continuity and differentiability.

At the conclusion of this semester you should be able to:

1. Understand what a real number is and what a metric space is.

2. Work with sequences and limits of sequences both for real numbers and general meteoric spaces.

3. Understand and work with the concepts of limits and continuity for functions again both for real numbers and general metric spaces.

4. Understand and work with the concept of differentiability and applications of differentiation such as the mean value theorem.

5. Be able to read mathematics and construct a rigorous mathematical argument.


Special Accommodations

Students who need special accomodations and who have documentation from Disability Services should make arrangements with me as soon as possible. Students should conact Disability Services for Students, Office of Student Life, 330 Memorial Union, 874-2098.