Name: Classical Descriptive Set Theory By Alexander Kechris
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Classical Descriptive Set Theory Summary

Classical Descriptive Set Theory by Alexander Kechris

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.

I Polish Spaces.- 1. Topological and Metric Spaces.- 1.A Topological Spaces.- 1.B Metric Spaces.- 2. Trees.- 2.A Basic Concepts.- 2.B Trees and Closed Sets.- 2.C Trees on Produtcs.- 2.D Leftmost Branches.- 2.E Well-founded Trees and Rank.- 2.F The Well-founded Part of a Tree.- 2.G The Kleene-Brouwer Ordering.- 3. Polish Spaces.- 3.A Definitions and Examples.- 3.B Extensions of Continuous Functions and Homeomorphisms.- 3.C Polish Subspaces of Polish Spaces.- 4. Compact Metrizable Spaces.- 4.A Basic Facts.- 4.B Examples.- 4.C A Universality Property of the Hilbert Cube.- 4.D Continuous Images of the Cantor Space.- 4.E The Space of Continuous Functions on a Compact Space.- 4.F The Hyperspace of Compact Sets.- 5. Locally Compact Spaces.- 6. Perfect Polish Spaces.- 6.A Embedding the Cantor Space in Perfect Polish Spaces.- 6.B The Cantor-Bendixson Theorem.- 6.C Cantor-Bendixson Derivatives and Ranks.- 7.Zero-dimensional Spaces.- 7.A Basic Facts.- 7.B A Topological Characterization of the Cantor Space.- 7.C A Topological Characterization of the Baire Space.- 7.D Zero-dimensional Spaces aa Subspaces of the Baire Space.- 7.F Polish Spaces as Continuous Images of the Baire Space.- 7.F Closed Subsets Homcomorphic to the Baire Space.- 8. Baire Category.- 8.A Meager Sets.- 8.B Baire Spaces.- 8.C Choquet Games and Spaces.- 8.D Strong Choquet Games and Spaces.- 8.E A Characterization of Polish Spaces.- 8.F Sets with the Baire Property.- 8.G Localization.- 8.H The Banach-Mazur Game.- 8.I Baire Measurable Functions.- 8.J Category Quantifiers.- 8.K The Kuratowski-Ulam Theorem.- 8.L Some Applications.- 8.M Separate and Joint Continuity.- 9. Polish Groups.- 9.A Metrizable and Polish Groups.- 9.B Examples of Polish Groups.- 9.C Basic Facts about Baire Groups and Their Actions.- 9.D Universal Polish Groups.- II Borel Sets.- 10. Measurable Spaces and Functions.- 10.A Sigma-Algebras and Their Generators.- 10.B Measurable Spaces and Functions.- 11. Borel Sets and Functions.- 11.A Borel Sets in Topological Spaces.- 11.B The Borel Hierarchy.- 11.C Borel Functions.- 12. Standard Borel Spaces.- 12.A Borel Sets and Functions in Separable Metrizable Spaces.- 12.B Standard Borel Spaces.- 12.C The Effros Borel Space.- 12.D An Application to Selectors.- 12.E Further Examples.- 12.F Standard Borel Groups.- 13. Borel Sets as Clopen Sets.- 13.A Turning Borel into Clopen Sets.- 13.B Other Representations of Borel Sets.- 13.C Turning Borel into Continuous Functions.- 14. Analytic Sets and the Separation Theorem.- 14.A Basic Facts about Analytic Sets.- 14.B The Lusin Separation Theorem.- 14.C Sousliri's Theorem.- 15. Borel Injections and Isomorphisms.- 15.A Borel Injective Images of Borel Sets.- 15.B The Isomorphism Theorem.- 15.C Homomorphisms of Sigma-Algebras Induced by Point Maps.- 15.D Some Applications to Group Actions.- 16. Borel Sets and Baire Category.- 16.A Borel Definability of Category Notions.- 16.B The Vaught Transforms.- 16.C Connections with Model Theory.- 16.D Connections with Cohen's Forcing Method.- 17. Borel Sets and Measures.- 17.A General Facts on Measures.- 17.B Borel Measures.- 17.C Regularity and Tightness of Measures.- 17.D Lusin's Theorem on Measurable Functions.- 17.E The Space of Probability Borel Measures.- 17.F The Isomorphism Theorem for Measures.- 18. Uniformization Theorems.- 18.A The Jankov, von Neumann Uniformization Theorem.- 18.B Large Section Uniformization Results.- 18.C Small Section Uniformization Results.- 18.D Selectors and Transversals.- 19. Partition Theorems.- 19.A Partitions with a Comeager or Non-meager Piece.- 19.B A Ramsey Theorem for Polish Spaces.- 19.C The Galvin-Prikry Theorem.- 19.D Ramsey Sets and the Ellentuck Topology.- 19.E An Application to Banach Space Theory.- 20. Borel Determinacy.- 20.A Infinite Games.- 20.B Determinacy of Closed Games.- 20.C Borel Determinacy.- 20.D Game Quantifiers.- 21. Games People Play.- 21.A The *-Games.- 21.B Unfolding.- 21.C The Banach-Mazur or **-Games.- 21.D The General Unfolded Banach-Mazur Games.- 21.E Wadge Games.- 21.F Separation Games and Hurewicz's Theorem.- 21.G Turing Degrees.- 22. The Borel Hierarchy.- 22. A Universal Sets.- 22.B The Borel versus the Wadge Hierarchy.- 22.C Structural Properties.- 22.D Additional Results.- 22.E The Difference Hierarchy.- 23. Some Examples.- 23.A Combinatorial Examples.- 23.B Classes of Compact Sets.- 23.C Sequence Spaces.- 23.D Classes of Continuous Functions.- 23.E Uniformly Convergent Sequences.- 23.F Some Universal Sets.- 23.G Further Examples.- 24. The Baire Hierarchy.- 24.A The Baire Classes of Functions.- 24.B Functions of Baire Class 1.- III Analytic Sets.- 25. Representations of Analytic Sets.- 25.A Review.- 25.B Analytic Sets in the Baire Space.- 25.C The Souslin Operation.- 25.D Wellordered Unions and Intersections of Borel Sets.- 25. E Analytic Sets as Open Sets in Strong Choquet Spaces.- 26. Universal and Complete Sets.- 26.A Universal Analytic Sets.- 26.B Analytic Determinacy.- 26.C Complete Analytic Sets.- 26.D Classification up to Borel Isomorphism.- 27. Examples.- 27.A The Class of Ill-founded Trees.- 27.B Classes of Closed Sets.- 27.C Classes of Structures in Model Theory.- 27.D Isomorphism.- 27.E Some Universal Sets.- 27.F Miscellanea.- 28. Separation Theorems.- 28.A The Lusin Separation Theorem Revisited.- 28.B The Novilcov Separation Theorem.- 28.C Borel Sets with Open or Closed Sections.- 28.D Some Special Separation Theorems.- 28.E Hurewicz-Type Separation Theorems.- 29. Regularity Properties.- 29.A The Perfect Set Property.- 29.B Measure. Category, and Ramsey.- 29.C A Closure Property for the Souslin Operation.- 29.D The Class of C-Sets.- 29.E Analyticity of Largeness Conditions on Analytic Sets.- 30. Capacities.- 30.A The Basic Concept.- 30.B Examples.- 30.C The Choquet Capacitability Theorem.- 31. Analytic Well-founded Relations.- 31.A Bounds on Ranks of Analytic Well-founded Relations.- 31.B The Kunen-Martin Theorem.- IV Co-Analytic Sets.- 32. Review.- 32.A Basic Facts.- 32.B Representations of Co-Analytic Sets.- 32.C Regularity Properties.- 33. Examples.- 33.A Well-founded Trees and Wellorderings.- 33.B Classes of Closed Sets.- 33.C Sigma-ldoals of Compact Sets.- 33.D Differentiable Functions.- 33.E Everywhere Convergence.- 33.F Parametrizing Baire Class 1 Functions.- 33.G A Method for Proving Completeness.- 33.H Singular Functions.- 33.I Topological Examples.- 33.J Homeomorphisms of Compact Spaces.- 33.K Classes of Separable Banach Spaces.- 33.L Other Examples.- 34. Co-Analytic Ranks.- 34.A Ranks and Prewellorderings.- 34.B Ranked Classes.- 34.C Co-Analytic Ranks.- 34.D Derivatives.- 34.E Co-Analytic Ranks Associated with Borel Derivatives.- 34.F Examples.- 35. Rank Theory.- 35.A Basic Properties of Ranked Classes.- 35.B Parametrizing Bi-Analytic and Borel Sets.- 35.C Reflection Theorems.- 35.D Boundedness Properties of Ranks.- 35.E The Rank Method.- 35.F The Strategic Uniformization Theorem.- 35.G Co-Analytic Families of Closed Sets and Their Sigma-Ideals.- 35.H Borel Sots with F? and K? Sections.- 36. Scales and Uniformiiatiou.- 36.A Kappa-Souslin Sets.- 36.B Scales.- 36.C Sealed Classes and Urniformization.- 36.D The Novikov-Kondo Uniformization Theorem.- 36.E Regularity Properties of Uniformizing Functions.- 36.F Uniforniizing Co-Analytic Sets with Large Sections.- 36.G Examples of Co-Analytic Scales.- V Projective Sets.- 37. The Projective Hierarchy.- 37.A Basic Facts.- 37.B Examples.- 38. Projective Determinacy.- 38.A The Second Level of the Projective Hierarchy.- 38.B Projective Determinacy.- 38.C Regularity Properties.- 39. The Periodicity Theorems.- 39.A Periodicity in the Projective Hierarchy.- 39.B The First Periodicity Theorem.- 39.C The Second Periodicity Theorem.- 39.D The Third Periodicity Theorem.- 40. Epilogue.- 40.A Extensions of the Projective Hierarchy.- 40.B Effective Descriptive Set Theory.- 40.C Large Cardinals.- 40.D Connections to Other Areas of Mathematics.- Appendix A. Ordinals and Cardinals.- Appendix B. Well-founded Relations.- Appendix C. On Logical Notation.- Notes and Hints.- References.- Symbols and Abbreviations.

Additional information

Sku

CIN0387943749G

ISBN 13

9780387943749

ISBN 10

0387943749

Title

Classical Descriptive Set Theory by Alexander Kechris

Author

Alexander Kechris

Series

Graduate Texts in Mathematics

Condition

Used - Good

Binding type

Hardback

Publisher

Springer-Verlag New York Inc.

Year published

1995-01-06

Number of pages

404

Prizes

N/A

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Book picture is for illustrative purposes only, actual binding, cover or edition may vary.

Note

This is a used book - there is no escaping the fact it has been read by someone else and it will show signs of wear and previous use. Overall we expect it to be in good condition, but if you are not entirely satisfied please get in touch with us

Classical Descriptive Set Theory Alexander Kechris

Classical Descriptive Set Theory by Alexander Kechris

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Classical Descriptive Set Theory Summary

Classical Descriptive Set Theory by Alexander Kechris

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