Primeless proofs of the Menger and Rothberger games

Abstract: We continue the study of the Menger and Rothberger games on lattices initiated in arXiv:2102.12901. This time, we extend the earlier results by dropping some hypotheses that turned out to be unnecessary, and use Stone duality to recover known game characterizations for dense open families. We also give a formulation of Ufin for partially ordered sets and prove its game characterization without any lattice assumption. Finally, almost disjoint families give complete distributive lattices on which the selection principles and the corresponding games differ. We obtain lower bounds $\operatorname{cov}(\mathcal M)$ and $\mathfrak d$ for the least sizes of such counterexamples.