Translation Results for Some Star-Selection Games

TL;DR

This paper investigates star-selection principles and games in topological spaces and hyperspaces, establishing new equivalences and connections with classical selection principles and cardinal invariants.

Contribution

It introduces a method to express star-selection principles as ordinary ones, extending existing results to Pixley-Roy hyperspaces and uniform spaces.

Findings

Star-selection principles can be written as ordinary selection principles.

Connections between star-Menger/Rothberger games and classical games are established.

New links between cardinal invariants are uncovered.

Abstract

We continue to explore the ways in which high-level topological connections arise from connections between fundamental features of the spaces, in this case focusing on star-selection principles in Pixley-Roy hyperspaces and uniform spaces. First, we find a way to write star-selection principles as ordinary selection principles, allowing us to apply our translation theorems to star-selection games. For Pixley-Roy hyperspaces, we are able to extend work of M. Sakai and connect the star-Menger/Rothberger games on the hyperspace to the $\omega$-Menger/Rothberger games on the ground space. Along the way, we uncover connections between cardinal invariants. For uniform spaces, we show that the star-Menger/Rothberger game played with uniform covers is equivalent to the Menger/Rothberger game played with uniform covers, reinforcing an observation of Lj. Ko\v{c}inac.