A note on a new ideal

Kwela, Adam

doi:10.1016/j.jmaa.2015.05.042

Cited by 18 publications

(15 citation statements)

References 22 publications

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“…A property of ideals can often be expressed by finding a critical ideal (in sense of some order on ideals) with respect to this property (see [22,Theorem 1.3], [23,Theorem 2] or [31,Theorems 2.1 and 3.3]). This approach is very effective, especially in the context of ideal convergence (see [25] or [26]).…”

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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Section: Preliminariesmentioning

confidence: 99%

Ideal weak QN-spaces

Kwela

2018

Topology and its Applications

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“…0 will be denoted by G. These ideals were later renamed as simple density ideals in [46,47]. Now the following result provides the first basic answer to our question and we can conclude that there are uncountably many distinct analytic P-ideals.…”

Section: How Many Distinct Analytic P-idealsmentioning

confidence: 85%

Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements

Das

2021

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“…in a slightly different waythe equivalence of the definition from [17] with the presented one is proved in [13]). (1) If an ideal I is ω-+-diagonalizable, then so is any ideal J ⊆ I.…”

Section: 1mentioning

confidence: 94%

“…Each ideal which is not dense, has to be weakly Ramsey and ω-+-diagonalizable.Proof. The first statement follows from Theorem 4.4 and the fact that the ideal WR is dense (cf [13,. Lemma 5.3]).…”

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confidence: 97%