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Transactions of the American Mathematical Society
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
Published electronically: August 31, 2007
MathSciNet review: 2346478
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-07-04243-2
PII: S 0002-9947(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