Completely nonmeasurable unions

• Autorzy: Robert Rałowski , Szymon Żeberski
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

Quasi-measurable cardinal; Nonmeasurable set; Bernstein set; c.c.c. ideal; Polish space

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 4
• Strony: 683-687

Daty

wydano

2010-08-01

online

2010-07-24

Twórcy

autor

Robert Rałowski

• Institute of Mathematics and Computer Science, Wrocław University of Technology, Wrocław, Poland,

autor

Szymon Żeberski

• Institute of Mathematics and Computer Science, Wrocław University of Technology, Wrocław, Poland,

Bibliografia

• [1] Brzuchowski J., Cichon J., Grzegorek E., Ryll-Nardzewski C., On the existence of nonmeasurable unions, Bull. Acad. Polon. Sci. Sér. Sci. Math., 1979, 27(6), 447–448
• [2] Cichon J., Morayne M., Rałowski R., Ryll-Nardzewski C., Żeberski S., On nonmeasurable unions, Topol. Appl., 2007, 154(4), 884–893 http://dx.doi.org/10.1016/j.topol.2006.09.013[Crossref]
• [3] Jech T., Set Theory, 3rd millenium ed., Springer, Berlin, 2003
• [4] Zeberski S., On completely nonmeasurable unions, MLQ Math. Log. Q., 2007, 53(1), 38–42 http://dx.doi.org/10.1002/malq.200610024[Crossref]

Identyfikatory

DOI

10.2478/s11533-010-0038-z