zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Classical descriptive set theory. (English) Zbl 0819.04002
Graduate Texts in Mathematics. 156. Berlin: Springer-Verlag. xx, 402 p. DM 79.00; öS 616.20; sFr 76.00 (1995).

The aim of this monograph is to give an introduction to classical descriptive set theory and ideas of connections with other areas of mathematics.

The core of the theory is contained in separate chapters on Borel, analytic, co-analytic and projective sets. Besides the recent knowledge of the classical theory on separation of sets, projections, uniformizations, selections etc., detailed study is devoted also to infinite games and their connections with definability of sets. The theory of ranks and scales is the central point in some parts of the theory of co-analytic and analytic sets. The periodicity theorems concerning properties of the higher projective classes form the main content of the last chapter.

There are many applications, and some closely connected topics are investigated, too. Let us point out, e.g., the very interesting part on connections with set theory including the forcing method, Ramsey type theory for sets with the Baire property and its application to the proof of Rosenthal’s theorem on Banach spaces containing l 1 . Extra paragraphs are devoted to measure theory and many results on hyperspaces of sets and σ-ideals of sets can be found.

The book contains in some directions much more than a pure introduction to the classical theory. Many classical results appear in a generality known only quite recently. There are many results and/or proofs due to the author.

The organization of the book is very clear and it can be easily used not only as an introductory text but also as a reference book for finding known results on topics the reader is interested in and for finding interesting applications. The proofs are often very brief and they presume some experience. There are about 400 exercises giving often more information about further known results with hints to prove them. The introductory chapter is devoted to Polish spaces and three appendices give some set-theoretical background needed throughout the book.

03E15Descriptive set theory (logic)
03-01Textbooks (mathematical logic)
28A05Classes of sets
03-02Research monographs (mathematical logic)
91A44Games involving topology or set theory
54H05Descriptive set theory (topological aspects)