Abstract

We investigate several ideal versions of the pseudointersection number $\mathfrak {p}$, ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant $\mathtt {cov}^*({\mathcal I})$ has a crucial influence on the studied notions. For an invariant $\mathfrak {p}_\mathrm {K}({\mathcal J})$ introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant $\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})$ introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathtt {cov}^*({\mathcal I})\}=\mathfrak {p},\qquad \min \{\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}\le \mathtt {cov}^*({\mathcal I}), \end{aligned}$$ respectively. In addition to the first inequality, for a slalom invariant $\mathfrak {sl_e}({\mathcal I},{\mathcal J})$ introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathfrak {sl_e}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}=\mathfrak {p}. \end{aligned}$$ Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished.