SPACES NOT DISTINGUISHING IDEAL CONVERGENCES OF REAL-VALUED FUNCTIONS, II

Abstract

In [13] we gave combinatorial characterizations of $\mathrm{non}\left(P\right)$ of spaces expressing non-distinguishability of some ideal convergences and semi-convergences of sequences of continuous functions. In the present paper we study three of these invariants: $\mathrm{non}\left(\right(I,JQN)-\mathrm{space})$, $\mathrm{none}\left(\right(I,{\le }_{\mathrm{K}}JQN)-\mathrm{space})$, and $\mathrm{none}\left(\mathrm{w}\right(I,JQN)-\mathrm{space})$. We study them in connection with partial orderings of ${}^{\omega }\omega$ restricted to relations between $I$-to-one functions and $J$-to-one functions. In particular we prove that $\mathrm{none}\left(\mathrm{w}\right(I,JQN)-\mathrm{space})\le \mathfrak{b}$ for every capacitous ideal $J$ on $\omega$. This generalizes the same result of Kwela for ideals $J$ contained in an $F_{\sigma }$-ideal. If $J$ is a capacitous $P$-ideal, then $\mathrm{non}\left(\right(I,JQN)-\mathrm{space})=\mathrm{none}\left(\right(I,{\le }_{\mathrm{K}}JQN)-\mathrm{space})=\mathfrak{b}$ for every ideal $I\subseteq J$ and $\mathrm{none}\left(\mathrm{w}\right(I,JQN)-\mathrm{space})=\mathfrak{b}$ for every ideal $I$ below $J$ in the Katĕtov partial quasi-ordering of ideals.