Families of Sets Without the Baire Property

Abstract

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 {Xi}1 i=1 of a topological space X which has an open subset homeomorphic to (Rn, ), n 1, where is some admissible extension of the Euclidean topology, such that the union of each non-empty proper subfamily of {Xi}1 i=1 does not have the Baire property in X. In the case when X is a separable metrizable manifold of finite dimension, each element of {Xi}1 i=1 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 {G \ U : U 2 V1(Q)} is a base for a topology (Q) on G. The space (G, (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 (Q1) * (Q2) (resp. (Q1) (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. i