Primeless proofs of the Menger and Rothberger games

Abstract

We continue the study of the Menger and Rothberger games on lattices carried out in"Pointless proofs of the Menger and Rothberger games"(Topology Appl. 300 (2021), 107774). 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 $\mathsf U_{\mathrm{fin}}$ 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. Dias's results give the lower bounds $\operatorname{cov}(\mathcal M)$ and $\mathfrak d$ for the least sizes of such counterexamples.