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On the families of sets without the Baire property generated by the Vitali sets
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Department of MathematicsNational University of RwandaButareRwanda.
2011 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 3, no 2, p. 100-107Article in journal (Refereed) Published
##### Abstract [en]

Let A be the family of all meager sets of the real line ℝ, V be the family of all Vitali sets of ℝ, V1be the family of all finite unions of elements of V and V2 = {(C \ A1) ∪ A2: C ∈ V1; A1, A2 ∈ A}. We show that the families VV1, V2 are invariant under translations of ℝ, and V1, V2 are abelian semigroups with the respect to the operation of union of sets. Moreover, V ⊂ V1 ⊂ V2 and V2consists of zero-dimensional sets without the Baire property. Then we extend the results above to the Euclidean spaces ℝnn ≥ 2, and their products with the finite powers of the Sorgenfrey line.

##### Place, publisher, year, edition, pages
Springer, 2011. Vol. 3, no 2, p. 100-107
##### Keyword [en]
Vitali set of ℝn, Baire property, nonmeasurable set in the sense of Lebesgue, Sorgenfrey line
Mathematics
##### Identifiers
OAI: oai:DiVA.org:liu-136258DiVA: diva2:1086505
Available from: 2017-04-03 Created: 2017-04-03 Last updated: 2017-05-09Bibliographically approved
##### In thesis
1. Families of Sets Without the Baire Property
Open this publication in new window or tab >>Families of Sets Without the Baire Property
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

The family of sets with the Baire property of a topological space X, i.e., sets which differ from open sets by meager sets, has different nice properties, like being closed under countable unions and differences. On the other hand, the family of sets without the Baire property of X is, in general, not closed under finite unions and intersections. This thesis focuses on the algebraic set-theoretic aspect of the families of sets without the Baire property which are not empty. It is composed of an introduction and five papers.

In the first paper, we prove that the family of all subsets of ℝ of the form (C \ M) ∪ N, where C is a finite union of Vitali sets and M, N are meager, is closed under finite unions. It consists of sets without the Baire property and it is invariant under translations of ℝ. The results are extended to the space ℝn for n ≥ 2 and to products of ℝn with finite powers of the Sorgenfrey line.

In the second paper, we suggest a way to build a countable decomposition $\tiny\lbrace\text X_i\rbrace _{i=1}^\infty$ of a topological space X which has an open subset homeomorphic to (ℝn, τ), n ≥ 1, where τ is some admissible extension of the Euclidean topology, such that the union of each non-empty proper subfamily of $\tiny\lbrace\text X_i\rbrace _{i=1}^\infty$ does not have the Baire property in X. In the case when X is a separable metrizable manifold of finite dimension, each element of $\tiny\lbrace\text X_i\rbrace _{i=1}^\infty$ can be chosen dense and zero-dimensional.

In the third paper, we develop a theory of semigroups of sets with respect to the union of sets. The theory is applied to Vitali selectors of ℝ to construct diverse abelian semigroups of sets without the Baire property. It is shown that in the family of such semigroups there is no element which contains all others. This leads to a supersemigroup of sets without the Baire property which contains all these semigroups and which is invariant under translations of ℝ. All the considered semigroups are enlarged by the use of meager sets, and the construction is extended to Euclidean spaces ℝn for n ≥ 2.

In the fourth paper, we consider the family V1(Q) of all finite unions of Vitali selectors of a topological group G having a countable dense subgroup Q. It is shown that the collection $\tiny\lbrace\text G \setminus \text U : \text U \in \ \mathcal{V}_1(\text{Q})\rbrace$ is a base for a topology τ(Q) on G. The space (G, τ (Q)) is T1, not Hausdorff and hyperconnected. It is proved that if Q1 and Q2 are countable dense subgroups of G such that Q1 ⊆ Q2 and the factor group Q2/Q1 is infinite (resp. finite) then τ(Q1$\tiny\not\subseteq$ τ(Q2) (resp. τ (Q1) ⊆ τ (Q2)). Nevertheless, we prove that all spaces constructed in this manner are homeomorphic.

In the fifth paper, we investigate the relationship (inclusion or equality) between the families of sets with the Baire property for different topologies on the same underlying set. We also present some applications of the local function defined by the Euclidean topology on R and the ideal of meager sets there.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1825
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:liu:diva-137074 (URN)10.3384/diss.diva-137074 (DOI)9789176855928 (ISBN)
##### Public defence
2017-05-30, Hörsal C:3, Campus Valla, Linköping, 10:15 (English)
##### Supervisors
Available from: 2017-05-05 Created: 2017-05-04 Last updated: 2017-05-22Bibliographically approved

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P-Adic Numbers, Ultrametric Analysis, and Applications
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Cite
Citation style
• apa
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