Displaying similar documents to “Two point sets with additional properties”

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.

Completely nonmeasurable unions

Robert Rałowski, Szymon Żeberski (2010)

Open Mathematics

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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...

Some topological properties of ω -covering sets

Andrzej Nowik (2000)

Czechoslovak Mathematical Journal

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We prove the following theorems: There exists an ω -covering with the property s 0 . Under c o v ( 𝒩 ) = there exists X such that ∀ B ∈ ℬ o r [ B ∩ X is not an ω -covering or X ∖ B is not an ω -covering]. Also we characterize the property of being an ω -covering.

MAD families and the rationals

Michael Hrušák (2001)

Commentationes Mathematicae Universitatis Carolinae

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Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that 𝔟 = 𝔠 implies that there is a Cohen indestructible MAD family. It follows that a Cohen indestructible MAD family is in fact indestructible by Sacks and Miller forcings. A...

Compact covering mappings and cofinal families of compact subsets of a Borel set

G. Debs, J. Saint Raymond (2001)

Fundamenta Mathematicae

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Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement " ∀ α ∈ ω ω , ω ₁ L ( α ) < ω ₁ "; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets...