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On Meager Additive and Null Additive Sets in the Cantor Space 2 ω and in ℝ

Tomasz Weiss (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Let T be the standard Cantor-Lebesgue function that maps the Cantor space 2 ω onto the unit interval ⟨0,1⟩. We prove within ZFC that for every X 2 ω , X is meager additive in 2 ω iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in 2 ω and ℝ.

On nonmeasurable images

Robert Rałowski, Szymon Żeberski (2010)

Czechoslovak Mathematical Journal

Let ( X , 𝕀 ) be a Polish ideal space and let T be any set. We show that under some conditions on a relation R T 2 × X it is possible to find a set A T such that R ( A 2 ) is completely 𝕀 -nonmeasurable, i.e, it is 𝕀 -nonmeasurable in every positive Borel set. We also obtain such a set A T simultaneously for continuum many relations ( R α ) α < 2 ω . Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.

On open maps of Borel sets

A. Ostrovsky (1995)

Fundamenta Mathematicae

We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers ℙ not G δ · F σ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ ℙ of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. The answer to the question as stated is negative if X is Baire but Y is not Baire.

On some properties of squares of Sierpiński sets

Andrzej Nowik (2004)

Colloquium Mathematicae

We investigate some geometrical properties of squares of special Sierpiński sets. In particular, we prove that (under CH) there exists a Sierpiński set S and a function p: S → S such that the images of the graph of this function under π'(⟨x,y⟩) = x - y and π''(⟨x,y⟩) = x + y are both Lusin sets.

On strong measure zero subsets of κ 2

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

We study the generalized Cantor space κ 2 and the generalized Baire space κ κ as analogues of the classical Cantor and Baire spaces. We equip κ κ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of κ 2 . We prove for successor κ = κ < κ that the ideal of strong measure zero sets of κ 2 is κ -additive, where κ is the size of the smallest unbounded family in κ κ , and that the generalized Borel conjecture...

On the bounding, splitting, and distributivity numbers

Alan S. Dow, Saharon Shelah (2023)

Commentationes Mathematicae Universitatis Carolinae

The cardinal invariants 𝔥 , 𝔟 , 𝔰 of 𝒫 ( ω ) are known to satisfy that ω 1 𝔥 min { 𝔟 , 𝔰 } . We prove that all inequalities can be strict. We also introduce a new upper bound for 𝔥 and show that it can be less than 𝔰 . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).

On the complexity of subspaces of S ω

Carlos Uzcátegui (2003)

Fundamenta Mathematicae

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set 2 X (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space S ω is F σ δ . In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of S ω . We show that S ω has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology. Moreover,...

On the complexity of sums of Dirichlet measures

Sylvain Kahane (1993)

Annales de l'institut Fourier

Let M be the set of all Dirichlet measures on the unit circle. We prove that M + M is a non Borel analytic set for the weak* topology and that M + M is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates M + M from D (or even L 0 ) , the set of all measures singular with respect to every measure in M . This extends results of Kaufman, Kechris and Lyons about D and H and gives many examples of non Borel analytic sets.

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