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A Course on Borel Sets

The roots of Borel sets go back to the work of Baire [8]. He was trying to come to grips with the abstract notion of a function introduced by Dirich­ let and Riemann. According to them, a function was to be an arbitrary correspondence between objects without giving any method or procedure by which t...

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Bibliographic Details
Main Author: Srivastava, S.M.
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1998, 1998
Edition:1st ed. 1998
Series:Graduate Texts in Mathematics
Subjects:
Mathematical Logic
Topology
Mathematical Logic And Foundations
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 1 Cardinal and Ordinal Numbers
  • 1.1 Countable Sets
  • 1.2 Order of Infinity
  • 1.3 The Axiom of Choice
  • 1.4 More on Equinumerosity
  • 1.5 Arithmetic of Cardinal Numbers
  • 1.6 Well-Ordered Sets
  • 1.7 Transfinite Induction
  • 1.8 Ordinal Numbers
  • 1.9 Alephs
  • 1.10 Trees
  • 1.11 Induction on Trees
  • 1.12 The Souslin Operation
  • 1.13 Idempotence of the Souslin Operation
  • 2 Topological Preliminaries
  • 2.1 Metric Spaces
  • 2.2 Polish Spaces
  • 2.3 Compact Metric Spaces
  • 2.4 More Examples
  • 2.5 The Baire Category Theorem
  • 2.6 Transfer Theorems
  • 3 Standard Borel Spaces
  • 3.1 Measurable Sets and Functions
  • 3.2 Borel-Generated Topologies
  • 3.3 The Borel Isomorphism Theorem
  • 3.4 Measures
  • 3.5 Category
  • 3.6 Borel Pointclasses
  • 4 Analytic and Coanalytic Sets
  • 4.1 Projective Sets
  • 4.2 ?11 and ?11 Complete Sets
  • 4.3 Regularity Properties
  • 4.4 The First Separation Theorem
  • 4.5 One-to-One Borel Functions
  • 4.6 The Generalized First Separation Theorem
  • 4.7 Borel Sets with Compact Sections
  • 4.8 Polish Groups
  • 4.9 Reduction Theorems
  • 4.10 Choquet Capacitability Theorem
  • 4.11 The Second Separation Theorem
  • 4.12 Countable-to-One Borel Functions
  • 5 Selection and Uniformization Theorems
  • 5.1 Preliminaries
  • 5.2 Kuratowski and Ryll-Nardzewski’s Theorem
  • 5.3 Dubins — Savage Selection Theorems
  • 5.4 Partitions into Closed Sets
  • 5.5 Von Neumann’s Theorem
  • 5.6 A Selection Theorem for Group Actions
  • 5.7 Borel Sets with Small Sections
  • 5.8 Borel Sets with Large Sections
  • 5.9 Partitions into G? Sets
  • 5.10 Reflection Phenomenon
  • 5.11 Complementation in Borel Structures
  • 5.12 Borel Sets with ?-Compact Sections
  • 5.13 Topological Vaught Conjecture
  • 5.14 Uniformizing Coanalytic Sets
  • References

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