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 The answer was given independently by Cichoń ([1], unpublished) and Fremlin ([5]).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.
Abstract. In this paper we study a notion of a κ-covering in connection with Bernstein sets and other types of nonmeasurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings. Definitions and notationIn 1993 Carlson in his paper [3] introduced a notion of κ-coverings and used it for investigating whether some ideals are or are not κ-translatable. Later on κ-coverings were studied by other authors, e.g. Muthuvel (cf. [7]) and Nowik (cf.[8], [9]). In this paper we present new results on κ-coverings in connection with Bernstein sets. We also introduce two natural generalizations of the notion of κ-coverings, namely κ-S-coverings and κ-I-coverings.We use standard set-theoretical notation and terminology from [1]. Recall that the cardinality of the set of all real numbers R is denoted by c. The cardinality of a set A is denoted by |A|. If κ is a cardinal number then
Abstract. We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire space. IntroductionFor each infinite subset T of the set ω of all natural numbers let us denote by K(T ) the σ-ideal of meagre subsets of the space 2 T with the canonical product topology. By L(T ) we denote the σ-ideal of Lebesgue measure zero subsets of 2 T with respect to the canonical product measure.Notice that if T is a subset of ω then we can indentify the spaces 2 T × 2 ω\T and the Cantor space 2 ω using the canonical homeomorphism π T defined by π T (x) = (x|T, x|(ω \ T )). Directly from the definition of meagre sets it follows that if A ∈ K(T ), then A × 2 ω\T ∈ K(ω). The same observation is also true for the ideal L(ω), but it is evidently false for the σ-ideal of all countable subsets of the Cantor space. We call this property of the ideals K(ω) and L(ω) productivity.There are other natural productive σ-ideals of subsets of the Cantor space, e.g. the σ-ideal K(ω) ∩ L(ω). It is interesting that among them there exists the least productive σ-ideal which contains all points. We call this ideal S 2 . There exists also the least productive ideal of subsets of 2 ω , and we call it I 2 . These ideals have Borel bases but they do not satisfy the countable chain condition-there exists a family of continuum many pairwise disjoint Borel sets outside the ideal S 2 . The ideal I 2 is not σ-additive, and the ideal S 2 is precisely σ-additive. The minimum cardinality of bases of these ideals is continuum. The covering number cov of both ideals is equal to the reaping cardinal r, and the last basic combinatorial invariants (cardinal numbers non) are described in terms of splitting cardinal numbers. These results show that both splitting and reaping cardinals are closely connected with natural mathematical objects on the classical Cantor space.We also consider the minimal productive σ-ideal S ω of subsets of the Baire space
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