Selection games and the Vietoris space

Abstract

We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as the well-known relationship between ω-covers of X and ordinary open covers of the finite powers of X. The primary utility of this method is to establish similar relationships with k-covers and the Vietoris space of compact subsets. Particularly, we show that some commonly studied selection principles are equivalent to a related hyperspace being Menger or Rothberger. We then apply these equivalences to correct a flawed argument in a previous paper which attempted to show that Hurewicz/Pawlikowski theorems are true for k-covers.