On the families of sets without the Baire property generated by the Vitali sets |
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| Authors: | Vitalij A Chatyrko Venuste Nyagaharwa |
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| Institution: | 1.Department of Mathematics,Linkoping University,Linkoping,Sweden;2.Department of Mathematics,National University of Rwanda,Butare,Rwanda |
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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: C ∈ V
1; A
1, A
2 ∈ A}. 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, V ⊂ V
1 ⊂ V
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. |
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| Keywords: | |
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