Sets with the Baire property in topologies defined from Vitali selectors on the real line

Let V be a Vitali selector related to a countable dense subgroup Q of the additive topological group (R, +) of real numbers.In this paper, we discuss various properties of the space (R, τ (V )), where τ (V ) is a topology having the collection {V +q : q ∈ Q} as a base.We prove that if W is a Vitali selector related to a countable dense subgroup P of (R, +), distinct from Q, then the spaces (R, τ (V )) and (R, τ (W )) are homeomorphic.As a consequence, the family of sets having the Baire property in (R, τ (V )) and the family of sets with the Baire property in (R, τ (W )) are Baire congruent; that is, there exist bijective mappingswhich preserve the Baire property.We further examine the Baire congruence relation between the families of sets with the Baire property in * -topologies derived from τ (V ) by using ideals of sets in the sense of Janković and Hamlett.