In this paper we describe a family of open questions concerning strong completeness properties associated with the Baire Category Theorem. Some of our questions deal with classical completeness topics such as de Groot’s subcompactness property and the property now called Choquet completeness, while others ask about more recent topics such as domain-representability and its relation to classical...

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a “good” additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are consid...

We show that a real sequence x is convergent if and only there exist regular matrix A an Fσδ-ideal I on N such the set of subsequences y for which Ay I-convergent second Baire category. This includes cases where ideal asymptotic density zero sets, Banach finite sets. The latter recovers old result given by Keogh Petersen in (1958) [26]. Our proofs are different nature rely recent results contex...

We de…ne combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey prope...

For any countable group satisfying the \weak Rohlin property", and for each dynamical property, the set of-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice Z d ; indeed, all countable discrete amenable groups. For an arbitrary countable group, let A be the set of-actions on the unit circle Y. We establish an Equival...

The following discusses a generalization of Baire category in a continuous set. The objective is to provide a meaningful classification of subsets of a continuous set as “large” or “small” sets in linearly ordered continuous sets. In particular, for cardinal number κ, the continuous ordered set 2∗ a subset of the set of dyadic sequences of length κ is discussed. We establish that this space, an...