On Large Covers and the Strong Closed Discrete Game

Abstract

We strengthen a recent selection game equivalence proved by L. Chiozini involving spaces and their associated spaces of continuous real-valued functions. This strengthening establishes that the Rothberger game is perfect- and Markov-information dual to the strong closed discrete game on the space of real-valued continuous functions. In the process, we extend the theory of strategic equivalences for the second player in variations of the Menger and Rothberger games involving large covers while noting explicitly how large covers present serious obstacles to applying strategy translation results used by the authors in similar scenarios in previous papers. We also briefly comment on subbasic selection games introduced by D. Guerrero S\'{a}nchez and V.V. Tkachuk.