On ideal equal convergence II

Staniszewski, Marcin

doi:10.1016/j.jmaa.2017.01.031

Cited by 12 publications

(9 citation statements)

References 15 publications

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“…The ideal version of quasi-normal convergence has been intensively studied e.g. in [11], [12], [16], [17], [26] and [32].…”

Section: Preliminariesmentioning

confidence: 99%

Ideal weak QN-spaces

Kwela

2018

Topology and its Applications

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This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics.Recently,Šupina in [33] proved that I is not a weak P-ideal if and only if any topological space is an IQN-space. Moreover, under p = c he constructed a maximal ideal I (which is not a weak P-ideal) for which the notions of IQNspace and QN-space do not coincide. In this paper we show that, consistently, there is an ideal I (which is not a weak P-ideal) for which the notions of IwQNspace and wQN-space do not coincide. This is a partial solution to [6, Problem 3.7]. We also prove that for this ideal the ideal version of Scheepers Conjecture does not hold (this is the first known example of such weak P-ideal).We obtain a strictly combinatorial characterization of non(IwQN-space) similar to the one given in [33] byŠupina in the case of non(IQN-space). We calculate non(IQN-space) and non(IwQN-space) for some weak P-ideals. Namely, we show that b ≤ non(IQN-space) ≤ non(IwQN-space) ≤ d for every weak P-ideal I and that non(IQN-space) = non(IwQN-space) = b for every Fσ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for b(I, I, Fin) introduced in [32]). As a consequence, we obtain some bounds for add(IQN-space). In particular, we get add(IQN-space) = b for analytic P-ideals I generated by unbounded submeasures.By a result of Bukovský, Das andŠupina from [6] it is known that in the case of tall ideals I the notions of IQN-space (IwQN-space) and QN-space (wQN-space) cannot be distinguished. Answering [6, Problem 3.2], we prove that if I is a tall ideal and X is a topological space of cardinality less than cov * (I), then X is an IwQN-space if and only if it is a wQN-space.

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“…The ideal version of quasi-normal convergence has been intensively studied e.g. in [11], [12], [16], [17], [26] and [32].…”

Section: Preliminariesmentioning

confidence: 99%

Ideal weak QN-spaces

Kwela

2018

Topology and its Applications

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“…Then, by Theorem 3.3, |A| < d, a contradiction. Remark 1º ω 1 < b(I) = c, b < b(I) = c, b < d = c and b ≤ b(I) < d = care all consistent with the axioms of (ZFC) (see[10,11,26]). Lemma 3.7, Theorem 3.3, Theorem 3.4, Theorem 3.5 and Theorem 3.6 are summarized in the following figure.…”

mentioning

confidence: 68%

The Absolutely Strongly Star-Hurewicz Property with Respect to an Ideal

Singh

Tyagi

Bhardwaj

2020

Tatra Mountains Mathematical Publications

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Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn, 𝒰n)}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

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“…In [9], the authors proved that the smallest size of non-QN-space is equal to the bounding number b. The cardinal numbers b(I, J, K) and b s (I, J , K) were introduced by Filip ów and Staniszewski [18,35] to characterize the smallest size of a space which is not ideal-QN. Recently, Repický [31,32] thoroughly examined ideal-QN spaces and, among others, characterized the smallest size of non-ideal-QNspaces in terms of the cardinal b(≥ I ∩(D K × D J )).…”

Section: The Bounding Numbers à La Staniszewskimentioning

confidence: 99%

“…The paper is organized in the following way. In Section 3 we show that b(> I ∩(D K × D J )) = b(I, J , K) and b(≥ I ∩(D K × D J )) = b s (I, J , K), where b(I, J , K) and b s (I, J , K) are cardinals considered by Filip ów and Stanszewski in [18,35]. This provides us with a very useful combinatorial characterizations of the considered cardinals, which we use almost exclusively in the rest of the paper.…”

Section: §1 Introduction For An Ideal I On We Denote D I = {F ∈mentioning

confidence: 99%

Yet Another Ideal Version of the Bounding Number

Filipów¹,

Kwela²

2021

J. symb. log.

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Let I be an ideal on . For f, g ∈ we write f ≤I g if f(n) ≤ g(n) for all n ∈ \ A with some A ∈ I. Moreover, we denote DI = {f ∈ : f -1 [{n}] ∈ I for every n ∈ } (in particular, D Fin denotes the family of all finite-to-one functions).We examine cardinal numbers b(≥I ∩(DI × DI )) and b(≥I ∩(D Fin × D Fin )) describing the smallest sizes of unbounded from below with respect to the order ≤I sets in D Fin and DI , respectively. For a maximal ideal I, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.We show that b(≥I ∩(D Fin × D Fin )) = b for all ideals I with the Baire property and that ℵ 1 ≤ b(≥I ∩(DI × DI )) ≤ b for all coanalytic weak P-ideals (this class contains all Π 0 4 ideals). What is more, we give examples of Borel (even Σ 0 2 ) ideals I with b(≥I ∩(DI × DI )) = b as well as with b(≥I ∩(DI × DI )) = ℵ 1 .We also study cardinals b(≥I ∩(DJ × DK)) describing the smallest sizes of sets in DK not bounded from below with respect to the preorder ≤I by any member of DJ. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN.

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On ideal equal convergence II

Cited by 12 publications

References 15 publications

Ideal weak QN-spaces

Ideal weak QN-spaces

The Absolutely Strongly Star-Hurewicz Property with Respect to an Ideal

Yet Another Ideal Version of the Bounding Number

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