Page 687 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 687
TOPOLOGY
Sets with the Baire Property in Topologies Defined From Vitali Selectors
of the Real Line
Venuste Nyagahakwa, venustino2005@yahoo.fr
University of Rwanda, Rwanda
Let F be the family of all countable dense subgroups of the additive topological group (R, +)
of real numbers, and let V be a Vitali selector related to some element Q of F. According
to the generalized version of Vitali’s theorem, it is well known that all elements of the family
P := {V + q : q ∈ Q} of translated copies of V by points of Q, do not have the Baire property
in R, with respect to the Euclidean topology, and they are not measurable in the Lebesgue sense.
In this paper, we consider the topological space (R, τ (V )), where τ (V ) is a topology having
P as a base. Apart from studying the topological properties of (R, τ (V )), we also look at the
relationship between the families of sets with the Baire property in topologies defined from
τ (V ), by using distinct ideals of sets on R. Moreover, we show that for any Qi ∈ F, the spaces
(R, τ (Vi)), i = 1, 2, where Vi is Vitali selector related to Qi, are homeomorphic. We further
prove that the families of sets with the Baire property in the spaces (R, τ (V1)) and (R, τ (V2))
are Baire congruent.
References
[1] A.B. Kharazishvili, Nonmeasurable sets and functions, Vol.195, Elsevier, 2004.
[2] D. Jankovic´ and T.R. Hamlett, New topologies from old via Ideals, Amer. Math. Monthly,
1990, Vol.97, No.4, 295 - 310.
[3] Vitalij A. Chatyrko and V. Nyagahakwa, Vitali selectors in topological groups and related
semigroups of sets, Questions And Answers in General Topology, 2015, Vol.33, No. 2, 93
- 102.
[4] Vitalij A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire prop-
erty generated by the Vitali sets, p-Adic Numbers, Ultrametric Analysis and Applications,
2011, Vol.3, No. 2, 100-107.
[5] G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905.
[6] J.C. Oxtoby, Measure and Category: Graduate texts in mathematics, Springer-Verlag,
New York, Heldenberg, Berlin, 1971.
685
Sets with the Baire Property in Topologies Defined From Vitali Selectors
of the Real Line
Venuste Nyagahakwa, venustino2005@yahoo.fr
University of Rwanda, Rwanda
Let F be the family of all countable dense subgroups of the additive topological group (R, +)
of real numbers, and let V be a Vitali selector related to some element Q of F. According
to the generalized version of Vitali’s theorem, it is well known that all elements of the family
P := {V + q : q ∈ Q} of translated copies of V by points of Q, do not have the Baire property
in R, with respect to the Euclidean topology, and they are not measurable in the Lebesgue sense.
In this paper, we consider the topological space (R, τ (V )), where τ (V ) is a topology having
P as a base. Apart from studying the topological properties of (R, τ (V )), we also look at the
relationship between the families of sets with the Baire property in topologies defined from
τ (V ), by using distinct ideals of sets on R. Moreover, we show that for any Qi ∈ F, the spaces
(R, τ (Vi)), i = 1, 2, where Vi is Vitali selector related to Qi, are homeomorphic. We further
prove that the families of sets with the Baire property in the spaces (R, τ (V1)) and (R, τ (V2))
are Baire congruent.
References
[1] A.B. Kharazishvili, Nonmeasurable sets and functions, Vol.195, Elsevier, 2004.
[2] D. Jankovic´ and T.R. Hamlett, New topologies from old via Ideals, Amer. Math. Monthly,
1990, Vol.97, No.4, 295 - 310.
[3] Vitalij A. Chatyrko and V. Nyagahakwa, Vitali selectors in topological groups and related
semigroups of sets, Questions And Answers in General Topology, 2015, Vol.33, No. 2, 93
- 102.
[4] Vitalij A. Chatyrko and V. Nyagahakwa, On the families of sets without the Baire prop-
erty generated by the Vitali sets, p-Adic Numbers, Ultrametric Analysis and Applications,
2011, Vol.3, No. 2, 100-107.
[5] G. Vitali, Sul problema della misura dei gruppi di punti di una retta, Bologna, 1905.
[6] J.C. Oxtoby, Measure and Category: Graduate texts in mathematics, Springer-Verlag,
New York, Heldenberg, Berlin, 1971.
685
