Investigations of ideal versions of selection principles pointed our attention to pseudointersection numbers introduced by P. Borodulin–Nadzieja and B. Farkas 2012 and M. Repický 2018. Both modifications of pseudointersection number are parametrized by ideals on natural numbers and related to orderings among such ideals.
We shall focus on these invariants and their connections to selection principles. Their topological characterizations (which resemble Fréchet-Urysohn property) are simple tool to derive inequalities among them. Finally, a result by P. Borodulin–Nadzieja and B. Farkas 2012 shows that ideal version of standard characterization of countable Fréchet-Urysohn property via selection principle by J. Gerlits and Zs. Nagy 1982 is not true.