Classical Descriptive Set Theory
Springer Science & Business Media, 6 груд. 2012 р. - 404 стор.
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.
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Perfect Polish Spaces
Standard Borel Spaces
Borel Sets as Clopen Sets
Borel Injections and Isomorphisms
Borel Sets and Measures
Scales and Uniformization
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algebra analytic sets assume Baire space Banach space bijection Borel function Borel measure Borel sets called Cantor set class of sets clearly clopen closed set closed subspace closed under continuous comeager compact metrizable compact sets Consider contains continuous function continuous preimages converges define denote equivalence relation Exercise finite function f given homeomorphic II-rank II'-complete infinite Kechris Lemma length(s Let f Let G Lusin meager measurable space metrizable space Moschovakis nonempty open sets notation open nbhd open sets ordinal pairwise disjoint player plays pointwise Polish group Polish space Pow(X Projective Determinacy projx Proof prove pruned tree rank recursion second countable separable Banach space sequence set A C X set theory sets in Polish Show space and A C X standard Borel space Theorem topological space transfinite uniformization unique Wadge well-founded well-founded relation winning strategy wins iff X-complete zero-dimensional