An Introduction to Set Theory and Topology

459 Pages·2016·7.22 MB·English

An Introduction to Set Theory and Topology
Ronald C. Freiwald
Washington University in St. Louis
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Copyright © 2014 Ronald C. Freiwald
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An Introduction to Set Theory and Topology

By Ronald C. Freiwald
2014
Washington University in St. Louis
Saint. Louis, Missouri

doi: 10.7936/K7D798QH
http://doi.dx.org/10.7936/K7D798QH
ISBN: 978-1-941823-10-1

https://creativecommons.org/licenses/by-nc-nd/4.0/

Cover design by Clayton Petras, MFA 2017, Sam Fox School of Design
& Visual Arts
Introduction
These notes are an introduction to set theory and topology. They are the result of teaching a two-
semester course sequence on these topics for many years at Washington University in St. Louis.
Typically the students were advanced undergraduate mathematics majors, a few beginning
graduate students in mathematics, and some graduate students from other areas that included
economics and engineering.
Over time my lecture notes evolved into written outlines for students, then written versions of the
more involved proofs. The full set of notes was a project completed during the years 2003-2007
with small revisions thereafter.
The usual background for the material is an introductory undergraduate analysis course, mostly
because it provides a solid introduction to Euclidean space  and practice with rigorous
argumentsin particular, about continuity. Strictly speaking, however, the material is mostly
self-contained. Examples are taken now and then from analysis, but they are not logically
necessary for the development of the material. The only real prerequisite is the level of
mathematical interest, maturity and patience needed to handle abstract ideas and to read and
write careful proofs. A few very capable students have taken this course before introductory
analysis (even, rarely, outstanding university freshmen) and invariably they have commented
later on how material eased their way into analysis.
The material on set theory is not done axiomatically. However, we do try to provide some
informal insights into why an axiomatization of the subject might be valuable and what some of
the most important results are. A student with a good grasp of the set-theoretic material
scattered throughout the notes, but heavily concentrated in Chapters I and VIIIwill know
all the informal set theory that most mathematicians ever need and will be in a strong position to
continue on to a study of axiomatic set theory.
The topological material is lies within the area traditionally labeled “general topology.” No
topics from algebraic topology are included. This was a conscious choice that reflects my own
training and tastes, as well as a conviction that students are usually rushed too quickly through
the basics of topology in order to get to “where the action is.” It is certainly true that general
topology has not been the scene of much research for several decades, and most of the research
that does still continue is closely related to set theory and mathematical logic. Nevertheless,
general topology contains a set of tools that most mathematicians need, whether for work in
analysis or other parts of topology.
Many of those basic tools (such as “compactness” and the “product topology”) seem very
abstract when a student first meets them. It takes time to develop an ownership of these tools.
This includes a sense of their significance, an appropriate “feel” for how they behave, and good
techniquein short, all the things necessary to make using a compactness argument, say, into a
completely routine tool. I believe this “absorption” process is often short-circuited in the rush to
move students along to algebraic topology. The result then can be an introduction to algebraic
topology where many tedious details are (appropriately) omitted and the student is ill-equipped
to fill them inor even to feel confident that the omissions are genuinely routine. When that
happens, a student can begin to feel that the subject has a vague, hand-waving quality about it.
These notes are designed to give the student the necessary practice and build up intuition. They
begin with the more concrete material (metric spaces) and move outward to the more general
ideas. The basic notions about topological spaces are introduced in the middle of the study of
metric spaces to illustrate the idea of increasing abstraction and to highlight some important
properties of metric spaces against a background where these properties fail. The result is an
exposition that is not as efficient as it could be if the more general definitions were stated in the
first place. In particular, many of the basic ideas about metric spaces (Chapter II) are revisited in
the introductory chapter on topological spaces (Chapter III).
Just as in any mathematics course, solving problems is essential. There are many exercises in the
notes, particularly in the early chapters. They vary in difficulty but it is fair to say that a majority
of the problems require some thought. Few, if any, could be genuinely called “trivial.” For
example, in Chapter I (Sets) there are no problems of the sort: prove that 
  . It is assumed that students are sufficiently sophisticated not to need that
sort of drill.
There are “Chapter Reviews” at the end of each chapter. A review consists of a list of
statements, each of which requires either an explanation or a counterexample. Presented with
statements whose truth is uncertain, students can develop confidence and intuition, learn to make
thoughtful connections and guesses, and build a tool chest of examples and counterexamples.
Nearly every review statement requires only an insight, a use of an earlier result in a new
situation, or the application of a more abstract result to a concrete situation. For almost every
true statement, an appropriate justification consists of at most a few sentences.
These notes were a “labor of love” over many years and are intended as an aid for students, not
as a work for publication. Such originality as there is lies in the selection of material and its
organization. Many proofs and exercises have been refashioned or polished, but others are more-
or-less standard fare drawn from sources some of which are now forgotten. Readers familiar
with the material will probably recognize overtones of my predecessors and contemporaries such
as Arthur H. Stone, Leonard Gillman, Robert McDowell and Stephen Willard. My thanks to
them for all their insights and contributions, and to a few hundred students who have worked
with various parts of these notes over the years. Of course, any errors are my own.
The notes are organized into ten chapters (I,II,...,X) and each chapter is divided into sections (1,
2,...,). Definitions, theorems, and examples are numbered consecutively within each of these
sectionsfor example, Definition 4.1, Theorem 4.2, Theorem 4.3, Example 4.4, .... For
example, a reference to Theorem 6.4 refers to the 4th numbered item in Section 6of the current
chapter. A reference to an item outside the current chapter would include the chapter number:
for example, Theorem III.6.4 means the 4th item in Section 6 of Chapter III.
Exercises are numbered consecutively within each chapter: E1, E2, ... . A reference to an
exercise outside the current chapter would include the chapter numberfor example, Exercise
III.E8.
Ronald C. Freiwald
St. Louis, Missouri
May 2014
Table of Contents
Chapter I Sets
1. Introduction 1
2. Preliminaries and Notation 2
3. Paradoxes 6
4. Elementary Operation on Sets 7
5. Functions 15
6. More About Functions 19
7. Infinite Sets 27
8. Two Mathematical Applications 36
9. More About Equivalent Sets 38
10. The Cantor-Schroeder-Bernstein Theorem 42
11. More About Subsets 45
12. Cardinal Numbers 47
13. Ordering the Cardinals 48
14. The Arithmetic of Cardinal Numbers 50
15. A Final Digression 58
Exercises 5, 13, 25,
32, 55
Chapter Review 59
Chapter II Pseudometric Spaces
1. Introduction 61
2. Metric and Pseudometric Spaces 61
3. The Topology of  70
4. Closed Sets and Operators on Sets 76
5. Continuity 85

Exercises 74, 82, 96
Chapter Review 99
Chapter III Topological Spaces
1. Introduction 103
2. Topological Spaces 103
3. Subspaces 109
4. Neighborhoods 110
5. Describing Topologies 113
6. Countability Properties of Spaces 126
7. More About Subspaces 131
8. Continuity 135
9. Sequences 142
10. Subsequences 146
Exercises 123, 129,
148
Chapter Review 151
Chapter IV Completeness and Compactness
1. Introduction 154
2. Complete Pseudometric Spaces 154
3. Subspaces of Complete Spaces 157
4. The Contraction Mapping Theorem 165
5. Completions 175
6. Category 178
7. Complete Metrizability 185
8. Compactness 193
9. Compactness and Completeness 200
10. The Cantor Set 203

Exercises 163, 174,
190, 207
Chapter Review 209
Chapter V Connected Spaces
1. Introduction 213
2. Connectedness 213
3. Path Connectedness and Local Path Connectedness 221
4. Components 225
5. Sierpinski's Theorem 229
Exercises 234
Chapter Review 237
Chapter VI Products and Quotients
1. Introduction 239
2. Infinite Products and the Product Topology 239
3. Productive Properties 251
4. Embedding Spaces in Products 261
5. The Quotient Topology 269

Exercises 248, 260,
266, 279
Chapter Review 281
Chapter VII Separation Axioms
1. Introduction 283
2. The Basic Ideas 283
3. Complete Regularity and Tychonoff Spaces 292
4. Normal and  -Spaces 302

5. Urysohn's Lemma and Tietze's Extension Theorem 304
6. Some Metrization Results 314
Exercises 290, 300,
316
Chapter Review 318
Chapter VIII Ordered Sets, Ordinals and Transfinite Methods
1. Introduction 319
2. Partially Ordered Sets 319
3. Chains 23
4. Order Types 331
5. Well-Ordered Sets and Ordinal Numbers 337
6. Indexing the Infinite Cardinals 350
7. Spaces of Ordinals 352
8. The Spaces  and   354
 
9. Transfinite Induction and Recursion 361
10. Using Transfinite Induction and Recursion 364
11. Zorn's Lemma 374
Appendix: Exponentiation of Ordinals 384
Exercises 330, 335, 359,
381
Chapter Review 386
Chapter IX Convergence
1. Introduction 388
2. Nets 389
3. Filters 394
4. The Relationship Between Nets and Filters 397
5. Ultrafilters and Universal Nets 401
6. Compactness Revisited and the
Tychonoff Product Theorem 405
7. Applications of the Tychonoff Product Theorem 409
Exercises 413
Chapter Review 416
Chapter X Compactifications
1. Basic Definitions and Examples 418
2. Local Compactness 420
3. The Size of Compactifications 423
4. Comparing Compactifications 425
5. The Stone-Cech Compactification 431
6. The Space  438
7. Alternate Constructions of  443
Exercises 447
Chapter Review 449