University of Rhode Island      Department of Mathematics

MTH 436  Real Analysis and Topology II

Monday, Wednesday 3 - 4:15pm Tyler Hall 106,
Friday 3 - 4pm Lippitt 204



Instructor  Dr. Mark Comerford
Office  Lippitt 102 F
Phone  874 5984
Email  mcomerford@math.uri.edu
Office Hours
  Thursday 2 - 5 pm
  or by appointment
Texts  Russell A. Gordon
Real Analysis - A First Course
(Second Edition),
 ISBN 0-201-43727-9
 Theodore W. Gamelin,
Robert Everist Greene
Introduction to Topology
(Second Edition),
 ISBN 0-486-40680-6
Prerequisites  Mth 435 or equivalent



Lectures and Homework Problems


Section Problems
3.5 Monotone Functions 6, 26
V   Topological Spaces Chapter V Problems
5.1 The Riemann Integral 15, 19
5.2 Conditions for Riemann Integrability 10, 12, 14
VI   Bases for a Topology
5.3 The Fundamental Theorem of Calculus 12, 13, 24
5.4 Further Properties of the Integral 3, 6, 25
VII  Interior, Closure, Boundary
5.5 Numerical Integration
6.1 Convergence of Infinite Series 3, 20
VIII Separation Axioms
6.2 The Comparison Tests 13, 21
6.3 Absolute Convergence 5, 14
IX  Compactness
6.4 Rearrangements and Products 3, 14
7.1 Pointwise Convergence 8, 9
X  More on Compactness
7.2 Uniform Convergence 7, 15
7.3 Uniform Convergence and Inherited Properties 3, 7
XI  Connectedness
7.4 Power Series 14, 16
7.5 Taylor's Formula
XII  Path Connectedness
7.6 Several Miscellaneous Results

 

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

Exams


Midterm  3 - 4:15pm, Monday March 4  Tyler 106
Final   3 - 6pm, Friday May 3   Tyler 106

 

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 a continuation of last semester's course on real analysis and topology. We will cover Chapters 5, 6 and 7 of Gordon's book and some extra topics, if there is time. We will also cover key concepts of point set topology from the basics of topological spaces to compactness and connectedness.

Once again, there will be strong emphasis on mathematical rigor and mastering techniques of proof. As with last semester, 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 March 4) 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 Riemann integration, infinite series and Taylor series and the beginnings of point set topology.

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

1. Understand the rigorous definition of the Riemann integral and be able to apply it.

2. Work with infinite series and be able to apply the well-known tests for convergence.

3. Understand and work with limits of functions, both pointwise and uniform.

4. Understand and work with Taylor series and Taylor's theorem.

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

6. Understand and be able to use fundamental concepts in point set topology. These include topological spaces, bases, interior, closure, boundary, compactness, connectedness and path connectedness.


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.