Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the unique representation of families of sets


Authors: Su Gao, Steve Jackson, Miklós Laczkovich and R. Daniel Mauldin
Journal: Trans. Amer. Math. Soc. 360 (2008), 939-958
MSC (2000): Primary 54H05, 22F05; Secondary 54E35, 03E75
DOI: https://doi.org/10.1090/S0002-9947-07-04243-2
Published electronically: August 31, 2007
MathSciNet review: 2346478
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ and $ Y$ be uncountable Polish spaces. $ A \subset X\times Y$ represents a family of sets $ \mathcal{C}$ provided each set in $ \mathcal{C}$ occurs as an $ x$-section of $ A$. We say that $ A$ uniquely represents $ \mathcal{C}$ provided each set in $ \mathcal{C}$ occurs exactly once as an $ x$-section of $ A$. $ A$ is universal for $ \mathcal{C}$ if every $ x$-section of $ A$ is in $ \mathcal{C}$. $ A$ is uniquely universal for $ \mathcal{C}$ if it is universal and uniquely represents $ \mathcal{C}$. We show that there is a Borel set in $ X\times R$ which uniquely represents the translates of $ \mathbb{Q}$ if and only if there is a $ \Sigma_2^1$ Vitali set. Assuming $ V = L$ there is a Borel set $ B \subset \omega^\omega$ with all sections $ F_\sigma$ sets and all non-empty $ K_\sigma$ sets are uniquely represented by $ B$. Assuming $ V =L$ there is a Borel set $ B \subset X\times Y$ with all sections $ K_\sigma$ which uniquely represents the countable subsets of $ Y$. There is an analytic set in $ X\times Y$ with all sections $ \Delta_2^0$ which represents all the $ \Delta_2^0$ subsets of $ Y$, but no Borel set can uniquely represent the $ \Delta_2^0$ sets. This last theorem is generalized to higher Borel classes.


References [Enhancements On Off] (What's this?)

  • 1. H. BECKER, Borel and Analytic One-One Parametrizations of the Countable Sets of Reals, Proc. of the A.M.S., Vol. 103, Number 3 (1988), 929-932. MR 947685 (89i:03091)
  • 2. H. BECKER AND A. S. KECHRIS, The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series 232, Cambridge University Press, Cambridge, 1996. MR 1425877 (98d:54068)
  • 3. D. CENZER AND R. D. MAULDIN, Inductive definability: Measure and category, Adv. in Math. 38 (1980), 55-90. MR 594994 (82b:03086)
  • 4. R. DOUGHERTY, S. JACKSON AND A. S. KECHRIS, The Structure of Hyperfinite Borel Equivalence Relations, Transactions of the A.M.S., Vol. 341, Number 1 (1994), 193-225. MR 1149121 (94c:03066)
  • 5. H. FRIEDMAN AND L. STANLEY, A Borel Reducibility Theory for Classes of Countable Structures, Journal of Symbolic Logic, Vol. 54 (1989) 894-914. MR 1011177 (91f:03062)
  • 6. W. HUREWICZ, Zur theorie der Analytischen Mengen, Fundamenta Mathematicae, vol. 15 (1930), 4-17.
  • 7. A. S. KECHRIS, Classical Descriptive Set Theory, Graduate Texts in Mathematics 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • 8. A. S. KECHRIS AND D. A. MARTIN, A Note on Universal Sets for Classes of Countable $ G_\delta$'s, Mathematika 22 (1975), 43-45. MR 0420579 (54:8593)
  • 9. K. KURATOWSKI, Topology. Volume 1, Academic Press, New York and London, 1966. MR 0217751 (36:840)
  • 10. A. LOUVEAU, A Separation Theorem for $ \Sigma^1_1$ Sets, Transactions of the A.M.S., Vol. 260, Number 2 (1980), 363-378. MR 574785 (81j:04001)
  • 11. R. D. MAULDIN, The boundedness of the Cantor-Bendixson order of some analytic sets, Pacific J. Math. 74 (1978), 167-177. MR 0474236 (57:13883)
  • 12. S. MAZURKIEWICZ AND W. SIERPINSKI, Sur un Problème Concernant les Fonctions Continues, Fundamenta Mathematicae, vol. 6 (1924), 161-169.
  • 13. Y. N. MOSCHOVAKIS, Descriptive Set Theory, North-Holland Publ. Co., New York, 1980. MR 561709 (82e:03002)
  • 14. J. SAINT-RAYMOND, Boréliens à coupes $ K_\sigma,$ Bull. Soc. Math. France 104 (1976), no. 4, 389-400. MR 0433418 (55:6394)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 54H05, 22F05, 54E35, 03E75

Retrieve articles in all journals with MSC (2000): 54H05, 22F05, 54E35, 03E75


Additional Information

Su Gao
Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
Email: sgao@unt.edu

Steve Jackson
Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
Email: jackson@unt.edu

Miklós Laczkovich
Affiliation: Department of Analysis, Eötvös Loránd University, Budapest, Kecskeméti u. 10-12, Hungary 1053
Email: laczk@cs.elte.hu

R. Daniel Mauldin
Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
Email: mauldin@unt.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04243-2
Keywords: Unique representations, uniquely universal sets, Vitali sets, scattered sets
Received by editor(s): April 28, 2005
Received by editor(s) in revised form: December 28, 2005
Published electronically: August 31, 2007
Additional Notes: The second author was supported by NSF grant DMS 0097181
The third author thanks the mathematics department of UNT for supporting his research visit.
The fourth author was supported by NSF grant DMS 0400481.
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society