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Classical Descriptive Set Theory (Graduate Texts in Mathematics) (v. 156) Hardcover – January 26, 1995

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Classical Descriptive Set Theory (Graduate Texts in Mathematics) (v. 156) + Descriptive Set Theory (Studies in Logic and the Foundations of Mathematics, Vol. 100)
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Product Details

  • Series: Graduate Texts in Mathematics (Book 156)
  • Hardcover: 404 pages
  • Publisher: Springer; 1995 edition (January 26, 1995)
  • Language: English
  • ISBN-10: 0387943749
  • ISBN-13: 978-0387943749
  • Product Dimensions: 9.5 x 6.3 x 1 inches
  • Shipping Weight: 1.6 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Best Sellers Rank: #1,367,150 in Books (See Top 100 in Books)

Editorial Reviews


Overall, the general impression is very good and this book will become a standard reference for the field. The book contains material from many perspective concerning descriptive set theory. I highly recommend it to any reader. -- R. Daniel Mauldin, Journal of Symbolic Logic, December 1997

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17 of 17 people found the following review helpful By Nathan Oakes on July 11, 2005
Format: Hardcover
There is a bit of unintended humor in the preface: "This book is essentially self-contained. The only thing it requires is familiarity...with the basics of general topology, measure theory, and functional analysis, as well as the elements of set theory..."

He says the target is the beginning graduate. I would place it better as a 2nd-year grad course. The text is dense and moves fast. Readability is pretty low. He never introduces a topic with context or overview. Extensive references to the literature were deliberately left out, which I think is wrong since it is a textbook. On the plus side, it is sprinkled with many exercises. (BTW, this is one of those cases that make you wish Springer didn't make authors do their own typesetting.)

There are only three common texts for descriptive set theory: Kechris, Jech, and Moschovakis. Jech has less detail on Polish spaces, Borel sets, and co-analytic sets, so it is not really a substitute, but its conciseness is nice and it makes a good companion. Moschovakis was a big deal when it came out because it collected a lot of information for the first time. But I don't think it is so good in content or style that you should be concerned if you have only Kechris and Jech.
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9 of 10 people found the following review helpful By A.T. on May 7, 2005
Format: Hardcover
A truly outstanding reference for the purely classical aspects of descriptive set theory, it falls under Kelley's label, "What every young set theorist needs to know." It is not an easy book for the beginner as it is very concise and gives little motivation, but for the advanced student it is essential.

As a Ph.D. student in the field, hardly a day goes by where I don't look up something in this book. I'm buying a new copy since my old one is falling apart.
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Format: Hardcover
You don't need to be familiar with serious set theory to read parts of this book. In particular, you don't have to know anything about forcing, which if I hear it in a talk alerts me that I won't understand what follows. The material on trees, schemes, and games is probably best skipped over if you are only trying to dip into this book for material relevant to analysis.

Three universal spaces that are talked about are the Hilbert cube I^N, where I=[0,1] and N denotes the set of positive integers, the Baire space N^N, and the Cantor space 2^N. These are all Polish spaces, and various results are proved about embedding spaces into these and expressing spaces as continuous images of these. For example, any separable metrizable space is homeomorphic to a subspace of the Hilbert cube, and every Polish space is homeomorphic to a G-delta subspace of the Hilbert cube; every nonempty compact metrizable space is a continuous image of the Cantor space; and any Polish space that is zero-dimensional (has a basis of clopen sets) all of whose compact subsets have empty interior is homeomorphic to the Baire space (this is the Alexandrov-Urysohn theorem).

The material on Polish spaces and Borel measures is excellent and is what I have used the book for. This book merits a place on the shelf of anyone who does analysis, in the sense of functional analysis, harmonic analysis, ergodic theory, and probability theory. Most of the book is probably too set theoretic to be of interest to an analyst, but you can make good use of this book without reading that material.

I think that exercises in a mathematics book should be tools for the reader to make more material unconscious, and especially to become comfortable with unravelling complicated definitions.
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