Ideal weak QN-spaces

Abstract

This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics.Recently, Šupina in [32] 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 IQN-space 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 IwQN-space 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 [32] by Šupina 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 [31]). 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 and Šupina 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.