Alexander S. Kechris
Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.
Not much alternative to this
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.
Your Review: Note: HTML is not translated!
Rating: Bad Good
Enter the code in the box below: