Pseudointersection numbers, ideal slaloms, topological spaces, and cardinal inequalities

Abstract

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