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On the unique representation of families of sets

Authors:	Su Gao Steve Jackson Mikló s Laczkovich R Daniel Mauldin

Institution:	Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203 ; Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203 ; Department of Analysis, Eötvös Loránd University, Budapest, Kecskeméti u. 10-12, Hungary 1053 ; Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203

Abstract:	Let and 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 -section of . We say that uniquely represents $\mathcal{C}$ provided each set in $\mathcal{C}$ occurs exactly once as an -section of . is universal for $\mathcal{C}$ if every -section of is in $\mathcal{C}$ . 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 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 . Assuming there is a Borel set $B \subset X\times Y$ with all sections $K_\sigma$ which uniquely represents the countable subsets of . There is an analytic set in $X\times Y$ with all sections $\Delta_2^0$ which represents all the $\Delta_2^0$ subsets of , but no Borel set can uniquely represent the $\Delta_2^0$ sets. This last theorem is generalized to higher Borel classes.

Keywords:	Unique representations uniquely universal sets Vitali sets scattered sets

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