Ideal weak QN-spaces

Abstract

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.

Yet Another Ideal Version of the Bounding Number

“…First we show that is a weak P-ideal (this fact is also shown in [22, Lemma 2.3], however we prove it here for the sake of completeness). Fix a partition .…”

“…Consider the ideal I = (Fin ⊗ Fin) ∩ (J ⊗ {∅}). First we show that I is a weak P-ideal (this fact is also shown in [22,Lemma 2.3], however we prove it here for the sake of completeness). To finish the proof we need to show that b(I, I, Fin) ≥ b(J , Fin, Fin).…”

“…A topological space X is a QN-space if it does not distinguish pointwise and quasi-normal convergence of sequences of real-valued continuous functions defined on X. QN-spaces were introduced by Bukovský et al in [9], and were thoroughly examined in papers [5,6,8,9,30,33,34,39]. The research on ideal-QN-spaces was initiated by Das and Chandra in [14] and has been continued in [7,22,34,36].…”

“…QN-spaces were introduced by Bukovský et al in [9], and were thoroughly examined in papers [5, 6, 8, 9, 30, 33, 34, 39]. The research on ideal-QN-spaces was initiated by Das and Chandra in [14] and has been continued in [7, 22, 34, 36].…”