Splittings

Kamburelis, Anastasis; W’glorz, B.

doi:10.1007/s001530050044

Cited by 27 publications

(18 citation statements)

References 5 publications

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“…The cardinal characteristic s almost , though employed only as a supporting actor in our proof below, may have some independent interest. Indeed, s almost is closely related to the finitely splitting number fs that was defined by Kamburelis and Węglorz in [7]. They proved that fs = max{b, s}.…”

Section: Claimmentioning

confidence: 94%

The subseries number

2019

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Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets? The answer to this question is defined to be the subseries number, a new cardinal characteristic of the continuum. This cardinal is bounded below by ℵ 1 and above by the cardinality of the continuum, but it is not provably equal to either. We define three natural variants of the subseries number, and compare them with each other, with their corresponding rearrangement numbers, and with several well-studied cardinal characteristics of the continuum. Many consistency results are obtained from these comparisons, and we obtain another by computing the value of the subseries number in the Laver model.

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Section: Claimmentioning

confidence: 94%

The subseries number

2019

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“…eglorz [4] that the least size of a block splitting family is max{b, s}. Therefore, when b ≤ s, there is a block splitting family of size s. Note that s n : n ∈ ω is a partition of ω into finite sets and that ∀i ∈ ω∀ ∞ n ∈ ω [s n ∩ a i = 0].…”

Section: It Was Proved By Kamburelis and W 'mentioning

confidence: 99%

Splitting Families and Complete Separability

Mildenberger¹,

Raghavan²,

Steprāns³

2014

Can. math. bull.

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“…It occurred that properties of this σ−ideal are strongly connected with some intensively studied combinatorial properties of subsets of natural numbers (the splitting and reaping numbers). For example, non(S 2 ) = ℵ 0 -s (see [11] for the definition of ℵ 0 -s and more discussion).…”

Section: Basic Definitions and Factsmentioning

confidence: 99%

Properties of ideals on the generalized Cantor spaces

Kraszewski

2001

J. symb. log.

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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.

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Splittings

Cited by 27 publications

References 5 publications

The subseries number

The subseries number

Splitting Families and Complete Separability

Properties of ideals on the generalized Cantor spaces

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