Properties of ideals on the generalized Cantor spaces

Abstract

Abstract. We define a class of productive σ−ideals of subsets of the Cantor space 2 ω and observe that both σ −ideals of meagre sets and of null sets are in this class. From every productive σ −ideal J we produce a σ −ideal J κ of subsets of the generalized Cantor space 2 κ . In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ , respectively. Then we investigate additivity, covering number, uniformity and cofinality of J κ . For example, we show thatOur results generalizes those from [5]. IntroductionIn this paper we shall discuss the properties of canonical σ−ideals of subsets of generalized Cantor spaces 2 κ , for example the σ −ideal of null sets and of meagre sets.In the 80's several people investigated relations between the σ −ideal N of null subsets of the classical Cantor space 2 ω and the σ − ideal N κ of null subsets of the generalized Cantor space 2 κ for some uncountable cardinal number κ. One of the most important questions was what were the connections between cardinal coefficients (such as add, cov, non and cof ) of N and these of N κ . The answer was given independently by Cichoń ([1], unpublished) and Fremlin ([5]). Both authors obtained almost the same results, except for two of them: Theorem 3.4 for null sets (only Fremlin) and Theorem 3.10 for null sets (only Cichoń).A natural question arose whether measure-theoretic tools were really necessary to get these results. In this paper we give a complete answer to it. In order to do this we extract the combinatorial principles that are considered by both authors and show that similar results to those which were obtained by them can be proved for a much wider class of ideals.In the first section we formulate a notion of productivity. If we identify 2 ω with its square using canonical homeomorphism then we can say a bit imprecisely that an ideal J of subsets of 2 ω is productive if for every A ∈ J the cylinder A × 2 ω is in J . We observe that σ −ideals of meagre sets and of null sets are productive.Received by the editors.

Transitive Properties of Ideals on Generalized Cantor Spaces

“…We investigate relations between transitive cardinal coefficients of J and those of J κ . Some of them are similar to relations between standard cardinal coefficients of J and J κ proved in [5]. We omit the proofs, as they are also analogous.…”

“…In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ , respectively. This description gives us a powerful tool for investigating combinatorial properties of ideals on 2 κ , which was done in [5]. In this paper we continue our research, focusing on transitive cardinal coefficients of ideals of subsets of 2 κ .…”

“…In 2001 Kraszewski in [5] defined a class of productive σ-ideals of subsets of the Cantor space 2 ω and observed that both σ-ideals of meagre sets and of null sets are in this class. Next, from every productive σ-ideal J one can produce a σ-ideal J κ of subsets of the generalized Cantor space 2 κ .…”

“…Let us also recall one useful definition used in [5]. We say that an ideal J of subsets of 2 ω has WFP (Weak Fubini Property) if for every ϕ ∈ Inj(ω, ω) and every A ⊆ 2…”

Productivity versus Weak Fubini Property

“…In 2001 Kraszewski in [4] defined a class of productive σ-ideals of subsets of the Cantor space 2 ω and observed that both σ-ideals of meagre sets M and of null sets N are in this class. Next, from every productive σ-ideal J one can produce a σ-ideal J κ of subsets of the generalized Cantor space 2 κ .…”