Selection Games and the Vietoris Space

TL;DR

This paper investigates the relationship between selection principles on Hausdorff spaces and their Vietoris hyperspaces, establishing equivalences that connect properties like Menger and Rothberger to hyperspace covers, and correcting previous errors in the literature.

Contribution

It introduces new connections between selection principles and Vietoris hyperspaces, extending known relationships to $k$-covers and fixing prior inaccuracies.

Findings

Selection principles on $X$ relate to properties of the Vietoris space.

Certain selection principles are equivalent to the hyperspace being Menger or Rothberger.

Corrects a flawed argument regarding $k$-covers and Pawlikowski's theorem.

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 $\omega$-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.