Displaying similar documents to “Completely nonmeasurable unions”

On nonmeasurable images

Robert Rałowski, Szymon Żeberski (2010)

Czechoslovak Mathematical Journal

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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 Bienias, Szymon Głąb, Robert Rałowski, Szymon Żeberski (2013)

Czechoslovak Mathematical Journal

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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...