Selection Games and the Vietoris Space

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 $\omega$-covers of $X$ and regular 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 a Pawlikowski theorem is true for $k$-covers.