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Tomek Bartoszynski

Publications and source records attributed to Tomek Bartoszynski.

At least 19 recordsLinked to original sources
arXiv2010

Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures

We consider the question, 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 is provably hereditary. This is contrasted with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes d$ and b, respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.

math.LO↗
arXiv2010

Additivity properties of topological diagonalizations

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions. This is a very concise paper. For a self-contained, complete, and extended treatment of the topic see math.GN/0604451

math.LO↗
arXiv2003

There may be no Hausdorff ultrafilters

An ultrafilter U is Hausdorff if for any two functions f,g mapping N to N, f(U)=g(U) iff f(n)=g(n) for n in some X in U. We will show that it is consistent that there are no Hausdorff ultrafilters.

math.LO↗
arXiv2003

Hechler's theorem for the meager ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing.

math.LO↗
arXiv2001

Strongly meager sets can be quite big

The paper contains two results pointing to the lack of symmetry between measure and category. Assume CH. There exists a strongly meager subset of the Cantor set that can be mapped onto the Cantor set by a uniformly continuous function. (It is well known that uniformly continuous image of a strongly null set is strongly null). A ZFC version of this result is also given.

math.LO↗
arXiv2001

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \aleph_1$ and find some topological characterizations of perfectly meager and universally meager sets.

math.LO↗
arXiv2001

On a theorem of Banach and Kuratowski and K-Lusin sets

In a paper of 1929, Banach and Kuratowski proved, assuming the continuum hypothesis, a combinatorial theorem which implies that there is no non-vanishing sigma-additive finite measure on the real line which is defined for every set of reals. It will be shown that the combinatorial theorem is equivalent to the existence of a K-Lusin set of size the continuum and that the existence of such sets is independent of ZFC plus non CH.

math.LO↗
arXiv2000

On perfectly meager sets

A set of reals A is called perfectly meager if A \cap P is meager in P, for every perfect set P. Marczewski asked if the product of perfectly meager sets is perfectly meager. In the paper it is shown that it is consistent that the answer to this question is positive. (It is known that it is also consistent that the answer is negative (Reclaw))

math.LO↗
arXiv2000

Continuous images of sets of reals

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.

math.LO↗
arXiv2000

Strongly meager sets do not form an ideal

A set X subseteq R is strongly meager if for every measure zero set H, X+H not= R. Let SM denote the collection of strongly meager sets. We show that assuming CH, SM is not an ideal.

math.LO↗
arXiv1999

Invariants of Measure and Category

A survey of recent results concerning cardinal invariants of measure and category. Submitted as a chapter of the upcoming Handbook of Set Theory.

math.LO↗
arXiv1999

Strongly meager and strong measure zero sets

We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).

math.LO↗
arXiv1999

Splitting number

It is shown that it is consistent with ZFC that every uncountable set can be continuously mapped onto a splitting family.

math.LO↗