On the families of sets without the Baire property generated by the Vitali sets

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Abstract

Let A be the family of all meager sets of the real line ℝ, V be the family of all Vitali sets of ℝ, V 1 be the family of all finite unions of elements of V and V 2 = {(C \ A 1) ∪ A 2: CV 1; A 1, A 2A}. We show that the families V, V 1, V 2 are invariant under translations of ℝ, and V 1, V 2 are abelian semigroups with the respect to the operation of union of sets. Moreover, VV 1V 2 and V 2 consists of zero-dimensional sets without the Baire property. Then we extend the results above to the Euclidean spaces ℝ n , n ≥ 2, and their products with the finite powers of the Sorgenfrey line.

Key words

Vitali set ofn Baire property nonmeasurable set in the sense of Lebesgue Sorgenfrey line 

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Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Department of MathematicsLinkoping UniversityLinkopingSweden
  2. 2.Department of MathematicsNational University of RwandaButareRwanda

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