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.
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