Ideal weak QN-spaces

Abstract

This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics. Recently, \v{S}upina proved that I is not a weak P-ideal if and only if any topological space is an IQN-space. Moreover, under $\mathfrak{p}=\mathfrak{c}$ 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. 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 ${\tt non}(\text{IwQN-space})$ similar to the one given by \v{S}upina in the case of ${\tt non}(\text{IQN-space})$. We calculate ${\tt non}(\text{IQN-space})$ and ${\tt non}(\text{IwQN-space})$ for some weak P-ideals. Namely, we show that $\mathfrak{b}\leq{\tt non}(\text{IQN-space})\leq{\tt non}(\text{IwQN-space})\leq\mathfrak{d}$ for every weak P-ideal I and that ${\tt non}(\text{IQN-space})={\tt non}(\text{IwQN-space})=\mathfrak{b}$ for every $\mathtt{F_\sigma}$ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for $\mathfrak{b}(I,I,Fin)$). As a consequence, we obtain some bounds for ${\tt add}(\text{IQN-space})$. In particular, we get ${\tt add}(\text{IQN-space})=\mathfrak{b}$ for analytic P-ideals I generated by an unbounded submeasure. By a result of Bukovsk\'y, Das and \v{S}upina 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. We prove that if I is a tall ideal and X is a topological space of cardinality less than ${\tt cov^*}(I)$, then X is an IwQN-space if and only if it is a wQN-space.