new to you, and begin reading right there. If things start to look new only around
Chapter 22 or so, you probably do not want to waste your time reading this book,
but go to the more advanced literature on the subject. In the "mathographical
remarks" at the end of most chapters, you will find ample suggestions for further
If things look new to you right from the beginning, check whether you know
most of the concepts and symbols listed under "Basic Notations". If so, read
Chapter 1, where some of the prerequisite material is reviewed. If Chapter 1 is
pleasant, easy reading, then you are probably ready for this book. If more than
two concepts listed under "Basic Notations" are entirely new to you, or if Chapter
1 feels challenging, then you may want to read one of the more elementary texts
listed in the mathographical remarks at the end of Chapter 1.
Roughly speaking, the only prerequisite for this book is that you are at ease
with set-theoretic notation. However, some knowledge of mathematical logic and
(for Volume II) general topology is indispensible. Therefore, to make the exposition
somewhat self-contained, we included a minicourse in mathematical logic in Chap-
ters 5 and 6, and also an Appendix on general topology at the end of Volume II.
Once you have determined your point of entrance, it is best to read the rest
of the book line by line. Much of this book is written like a dialogue between
the authors and the reader. This is intended to model the practice of creative
mathematical thinking, which more often than not takes on the form of an inner
dialogue in a mathematician's mind. You will quickly notice that this text contains
many question marks. This reflects our conviction that in the mathematical thought
process it is at least as important to have a knack for asking the right questions at
the right time as it is to know some of the answers.
You will benefit from this format only if you do your part and actively participate
in the dialogue. This means in particular: Whenever we pose a rhetorical question,
pause for a moment and ponder the question before you read our answer. Sometimes
we put a little more pressure on you and call our rhetorical questions EXERCISES.
Not all exercises are rhetorical questions that will be answered a few lines later.
Sometimes, the completion of a proof is left as an exercise. We also may ask you
to supply the entire proof of an interesting theorem, or an important example.
Nevertheless, we recommend that you attempt the exercises right away, especially
all the easier ones. Most of the time it will be easier to digest the ensuing text if
you have worked on the exercise, even if you were unable to solve it.
Here is a well-kept secret: All mathematical research papers contain plenty
of exercises. These usually appear under the disguise of seemingly unnecessary
assumptions, missing examples, or phrases like: "It is easy to see." One of the
most important steps in becoming a mathematician is to learn to recognize hidden
EXERCISES, and to develop the habit of tackling them right away.
We often make references to solutions of exercises from earlier chapters. Some-
times, the new material will make an old and originally quite hard exercise seem
trivial, and sometimes a new question can be answered by modifying the solution
to a previous problem. Therefore, it is a good idea to collect your solutions and
even your failed attempts at solutions in a folder where you can look them up later.
The level of difficulty of our exercises varies greatly. To help the reader save
time, we rated each exercise according to what we perceive as its level of difficulty.
The rating system is the same as used by American movie theatres. Everybody
should attempt the exercises rated G (general audience). Beginners are encouraged
Previous Page Next Page