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REMARKS ON THE SECOND EDITION
Many changes in, additions to, and some deletions from the first edition have been
made in accordance with thoughtful criticism from many colleagues at large and small
institutions.
A major feature of this new edition is the addition of sections called "Notes and
Additional Exercises," which include a variety of material. There are famous theorems
related to the material in the body of the text-for example, the Schroder-Bernstein
theorem from set theory, the Tietze extension theorem from topology, and Stone's
generalization of the Weierstrass approximation theorem. The proofs are given in outline
only, with a great deal left to the student as exercises. In these new sections there are also
miscellaneous exercises (many of which are quite challenging) and an occasional historical
note. An instructor's solutions manual for the problems in the new material can be
obtained from the author.
I also added an appendix that contains an axiomatic treatment of the real number
system. This was a compromise between no treatment at all in the first edition and a
lengthy development from basic principles that I think would retard the reader's progress
into the core of the book. All the assumptions about the real numbers and the necessary
results that can be derived from these assumptions are carefully presented.
There are a number of pictorial illustrations-also a departure from the first editionand
new exercises in many of the chapters, and new proofs.
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