Topology, Math 5520, Winter 2001

Graded homework can be picked up from the door of my office anytime Thursday or Friday, Feb 22 or 23, to help you study for the test Friday. Pass the word along to everyone else if you can, please.

Table of Contents

• Class data
• Professor
• Schedule
• Introduction
• Requirements and grading
• Books on reserve at the library
• Assignments

Class:

• Math 5520, Section 001
  
• 115 State Hall
  
• MWF    12:50 - 1:45
• Text:  Introduction to Topology, Crump W. Baker

Professor:

• Robert Bruner
  
• Office: 1201 FAB,   (313)-577-3179
• Math Department:   (313)-577-2479
• Fax:   (313)-577-7596 (shared)
• e-mail: rrb@math.wayne.edu
• Office hours: Mon & Wed 11:45-12:40; Wed 3:00 - 4:00.
  Short questions can often be answered after class. In addition, you are welcome to make appointments to see me at other times. If you are willing to take a chance that I am busy or not in, you can simply drop by my office. Finally, email is often a very effective means of getting in touch with me.

Schedule:

Here is a schedule of the topics and sections we will cover, as well as the timing of the tests.

Introduction	Chapter 1	1 week
Topological Spaces	Chapter 2	2 weeks
Subspaces and Continuity	Chapter 3	2 weeks
Products	Chapter 4	2 weeks
Midterm Exam	Friday, Feb. 23	(subject to change)
Connectedness	Chapter 5	2 weeks
Spring Break	Mar 12 - Mar 16
Compactness	Chapter 6	2 weeks
Separation	Chapter 7	1 week
Metric Spaces	Chapter 8	2 weeks
Final Exam	Monday, April 30	10:40 - 1:10

Introduction

Topology is one of the most fundamental ideas of modern mathematics. The simplicity of the ideas of topological space and continuous function are a wonderful illustration of the power of abstraction. They are the distillation of hundreds of years of experience. Their simplicity allows complete clarity in the derivation of their properties, without the need for vague intuitive arguments. This makes the course an ideal introduction to mathematical proof.

There are some short essays available on the web explaining what topology is in an intuitive way. Mine is What is Topology? . Neil Strickland, at the University of Sheffield has an essay of the same name which focuses on different aspects of the issue, and the algebraic topology discussion list run by Don Davis has an index of related essays.

For the historical context and issues the study of topology tries to answer, see the discussion for the first day.

Requirements and grades:

First, READ THE BOOK . You should read each section before we talk about it in class, then again after class, before doing the homework for the section. If you have any trouble understanding it, read it several times, first, quickly for an overall idea what the section is about, then in detail, working out the examples the book uses to make sure you know why each statement is true. Only after this should you start the homework. You may be pleasantly surprised how much easier the homework is with this sort of preparation. You will certainly understand the material and retain more of it, if you study in this way.

To help you in your reading, I will post discussion questions you should keep in mind while reading the text. You will be expected to give a brief answer to each either by email before class, or on paper at the start of class. These will be graded on a pass/fail basis, with any honest attempt to answer the questions counting as a pass.

Second, regular attendance and class participation will be expected. The only way to learn mathematics is to do it. Regular homework will be assigned and selected problems will be collected and graded at the start of each week.

I will expect all work to be written in clear English sentences and to exhibit correct logical arguments. Don't worry if this seems scary at the start: learning how to do this is one of the course's objectives. The use of scratch paper and rough drafts will be essential to accomplishing this, and I will be available to criticize rough drafts as needed.

Grades will be computed as follows:

Reserve books:

• Crump W. Baker, Introduction to Topology
  The text is on reserve so that if you need to consult it and don't have your copy handy, it will be available.
• John D. Baum, Elements of Point Set Topology
  This was my second choice for text: if Baker's book were not available I might well have used this one.
• Andrew Gleason, Fundamentals of Abstract Analysis
  If we have time, I will show you how to construct the real numbers starting just from the basics of set theory. Gleason's book goes through this in detail. It is also a good reference for the basics of set theory and logic which we shall use.
• Smith et al, Transition to Advanced Mathematics
  This book is devoted exclusively to the basic ideas about sets and proofs which we shall use. If you feel you need more help on these topics, this would be a good source.
• Stephen Willard, General Topology
  A comprehensive introduction to topology, which covers the topics in great depth. He provides a wealth of examples and couterexamples. It is written in a terse and elegant style which is well worth emulating.

Assignments