Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156) 1995th Edition by Alexander Kechris (PDF)

Description

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text presents a largely balanced approach to the subject, which combines many elements of the different traditions. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.

User’s Reviews

Editorial Reviews: Review 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 From the Back Cover Descriptive set theory is the area of mathematics concerned with the study of the structure of definable sets in Polish spaces. Beyond being a central part of contemporary set theory, the concepts and results of descriptive set theory are being used in diverse fields of mathematics, such as logic, combinatorics, topology, Banach space theory, real and harmonic analysis, potential theory, ergodic theory, operator algebras, and group representation theory. This book provides a basic first introduction to the subject at the beginning graduate level. It concentrates on the core classical aspects, but from a modern viewpoint, including many recent developments, like games and determinacy, and illustrates the general theory by numerous examples and applications to other areas of mathematics. The book, which is written in the style of informal lecture notes, consists of five chapters. The first contains the basic theory of Polish spaces and its standard tools, like Baire category. The second deals with the theory of Borel sets. Methods of infinite games figure prominently here as well as in subsequent chapters. The third chapter is devoted to the analytic sets and the fourth to the co-analytic sets, developing the machinery associated with ranks and scales. The final chapter gives an introduction to the projective sets, including the periodicity theorems. The book contains over four hundred exercises of varying degrees of difficulty.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A very handy introduction to the subject and the myriad applications and connections to other branches of mathematics, including logic. Recommended for graduate students and advanced undergraduates wih backgrounds in logic, analysis, or dynamic systems. However, Kechris is fairly terse, leaving many details to the reader.

⭐A masterful book. The theory of analytic sets is a gem. The book is written in a modular fashion, so you can get to the chapters that interest you.Just when you think you have reached the most subtle level of the Theory, Kechris shows you there is yet another level…

⭐I got this book because I had some lecture notes on Dr. Kechris’ class at CalTech and wanted more. It’s definitely worth the price. I learned a lot.

⭐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. For example, when working with metrizable spaces, assertions about compatible metrics producing the same topologies on objects that a priori depend on the metrics would be good exercises. I don’t think the exercises in this book are well chosen pedagogically; some of them are routine, but some (and these are not marked in any way) would certainly overwhelm a reader of ordinary skill. For example, one of the earliest questions in the book can be stated as proving that the irrationals between 0 and 1 are homeomorphic to the Baire space using the continued fraction expansion of an irrational number, and I expect that this is more likely to turn into a hopeless mess than it is to be an introduction to the fascination of continued fractions. If there are interesting results that Kechris wants to point out or to use later in the text, rather than pretending that they are good practice for a learner, he should call them “Remarks” and give a citation to the best written proof of the result he knows.

⭐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.

⭐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.

⭐I have no doubt about the talent of the author of this book, nor the fact that the text presents a wide range of results on descriptive set theory. But I found it very hard to follow the proofs and felt that the author did little to motivate the subject. This is a bit of a shame because descriptive set theory, and the descriptive theory of real numbers in particular, has produced some of the clearest thinkers of 19th and 20th Century mathematics: Rene Baire, Emile Borel, Henri Lebesgue, Nicholas Luzin and Waclaw Sierpinski (in no particular order). The upshot is that I use this book as a reference for theorems, and then go to other sources to find out why they are true. But it could just be me …

⭐I have used the book in research as a graduate student and a postdoc. It is a very good resource but, like many other mathematicians, Kechris has used a poor didactic style. The author plunges too fast into abstraction while I think it would be more effective to develop useful examples to motivate the theory. Some exercises are just not really ‘exercises’ but parts of research papers.

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