Ideal QN-spaces

Ideal weak QN-spaces

“…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] by Šupina in the case of non(IQN-space). We calculate non(IQN-space) and non(IwQN-space) for some weak P-ideals.…”

“…The fact that I is a weak P-ideal can be expressed equivalently by Fin⊗ Fin ≤ KB I (for this and other equivalent definitions of this notion, including the ones using different orders on ideals, see [33,Theorem 3.2]). If I is an ideal on ω, then we define the ideal I ⊗ ∅ on ω × ω, consisting of all A ⊆ ω × ω such that:…”

“…(Šupina,[33, Proposition 4.3]) Let I and J be two ideals on ω such that I ≤ KB J . Then any IQN-space is a J QN-space.…”

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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.

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Ideal approach to convergence in functional spaces

“…Note that Fin × Fin ≤ K J if and only if Fin × Fin ≤ 1-1 J , see [15]. For other equivalent conditions to Fin × Fin ≤ K J and further references, see [15,46,29]. Similarly, Fin α ≤ K J if and only if Fin α ≤ 1-1 J by [3].…”