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Bernstein sets with algebraic properties

100%

Kysiak M.

Open Mathematics

2009

tom 7

nr 4

725-731

We construct Bernstein sets in ℝ having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.

Completely nonmeasurable unions

88%

Rałowski R., Żeberski S.

Open Mathematics

2010

tom 8

nr 4

683-687

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 ⊆ $$ \mathbb{I} $$ of X contains uncountably many pairwise disjoint subfamilies , with $$ \mathbb{I} $$-Bernstein unions ∪ (a subset A ⊆ X is $$ \mathbb{I} $$-Bernstein if A and X \ A meet each Borel $$ \mathbb{I} $$-positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].

Ograniczanie wyników

2 Open Mathematics

1 Kysiak M.

1 Rałowski R.

1 Żeberski S.

1 2010

1 2009

Wyniki wyszukiwania

Bernstein sets with algebraic properties

Completely nonmeasurable unions