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Completely nonmeasurable unions

Robert RałowskiSzymon Żeberski — 2010

Open Mathematics

Assume that no cardinal κ < 2ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover ⊆ 𝕀 of X contains uncountably many pairwise disjoint subfamilies , with 𝕀 -Bernstein unions ∪ (a subset A ⊆ X is 𝕀 -Bernstein if A and X A meet each Borel 𝕀 -positive subset...

On nonmeasurable images

Robert RałowskiSzymon 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.

Two point sets with additional properties

Marek BieniasSzymon GłąbRobert RałowskiSzymon Żeberski — 2013

Czechoslovak Mathematical Journal

A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some σ -ideal, being (completely) nonmeasurable with respect to different σ -ideals, being a κ -covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider natural generalizations...

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