Tomek Bartoszynski

Set Theory: On the Structure of the Real Line

This research level monograph reflects the current state of the field and provides a reference for graduate students entering the field as well as for established researchers.

Hereditary topological diagonalizations and the Menger-Hurewicz conjectures

We consider the question of which of the major classes defined by topological diagonalizations of open or Borel covers is hereditary. Many of the classes in the open case are not hereditary already in ZFC, and none of them are provably hereditary. This is in contrast with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes 0 and b, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.

Additivity properties of topological diagonalizations

In a work of Just, Miller, Scheepers and Szeptycki it was asked whether certain diagonalization properties for sequences of open covers are provably closed under taking flnite or countable unions. In a recent work, Scheepers proved that one of the prop- erties in question is closed under taking countable unions. After surveying the known results, we show that none of the remain- ing classes is provably closed under taking flnite unions, and thus settle the problem. We also show that one of these properties is consistently (but not provably) closed under taking unions of size less than the continuum, by relating a combinatorial version of this problem to the Near Coherence of Filters (NCF) axiom, which as- serts that the Rudin-Keisler ordering is downward directed.

Continuous images of sets of reals

Abstract We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.

Closed measure zero sets

Abstract Bartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 (1992) 93–110. We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.

REMARKS ON SETS RELATED TO TRIGONOMETRIC SERIES

We show that several classes of sets, like N0-sets, Arbault sets, N-sets and pseudo-Dirichlet sets are closed under adding sets of small size.

All meager filters may be null

We show that in the Cohen model the sum of two nonmeasurable sets is always nonmeager. As a conseguence we show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property

On the cofinality of the smallest covering of the real line by meager sets. II

We study the ideal of meager sets and related ideals.

On perfectly meager sets

We show that it is consistent that the product of perfectly meager sets is perfectly meager.

Strongly Meager sets do not form an ideal

A set X⊆ℝ is strongly meager if for every measure zero set H, X+H ≠ℝ. Let