Ideal weak QN-spaces

Abstract

This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics. Recently, Šupina 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 $\mathtt{non}(\text{IwQN-space})$ similar to the one given by Šupina in the case of $\mathtt{non}(\text{IQN-space})$. We calculate $\mathtt{non}(\text{IQN-space})$ and $\mathtt{non}(\text{IwQN-space})$ for some weak P-ideals. Namely, we show that $\mathfrak{b}\le \mathtt{non}(\text{IQN-space})\le \mathtt{non}(\text{IwQN-space})\le \mathfrak{d}$ for every weak P-ideal I and that $\mathtt{non}(\text{IQN-space})=\mathtt{non}(\text{IwQN-space})=\mathfrak{b}$ 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 $\mathtt{add}(\text{IQN-space})$. In particular, we get $\mathtt{add}(\text{IQN-space})=\mathfrak{b}$ for analytic P-ideals I generated by an unbounded submeasure. By a result of Bukovský, Das and Šupina 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 $\mathtt{co}{\mathtt{v}}^{\ast }(I)$, then X is an IwQN-space if and only if it is a wQN-space.

References in corpus (4)

• More on cardinal invariants of analytic P-ideals
• Erdős-Ulam ideals vs. simple density ideals
• Ranks of $\mathcal{F}$-limits of filter sequences
• Ideal equal Baire classes

Cited by in corpus (3)

• Yet another ideal version of the bounding number
• On the structure of Borel ideals in-between the ideals $\mathcal{E}\mathcal{D}$ and $\mathrm{Fin}\otimes \mathrm{Fin}$ in the Katětov order
• More on yet another ideal version of the bounding number