Ideal QN-spaces

We continue to investigate an ideal version of QN-space, a JQN-space, introduced by P. Das and D. Chandra [8]. Following R. Filipów and M. Staniszewski [15], we show that an ideal J on ω contains an isomorphic copy of the ideal Fin×Fin on ω×ω if and only if every topological space is a JQN-space. If J does not contain an isomorphic copy of the ideal Fin×Fin then the Baire space ωω is not a JQN-space. However, if p=c then there is an ideal J not containing an isomorphic copy of the ideal Fin×Fin and there is a JQN-space which is not a QN-space. We prove few results related to an ideal version of an S1(Γ,Γ)-space. Indeed, we show that there is no ideal J such that the notion of an S1(Γ,J-Γ)-space is trivial. Consequently the ideal version of Scheepers' Conjecture does not hold for ideals containing an isomorphic copy of Fin×Fin.