Selection Games on Hyperspaces

TL;DR

This paper explores the relationship between selection principles and games on topological spaces and their hyperspaces, generalizing existing results using new techniques across various topologies and strategies.

Contribution

It introduces a novel technique to extend connections between topological constructs and selection games, applicable to various game lengths, strategies, and hyperspace topologies.

Findings

Unified selection principles for hyperspaces with Vietoris and Fell topologies

Extended connections between topological properties and selection games

Applicable to strategies of different strengths and game lengths

Abstract

In this paper we connect selection principles on a topological space to corresponding selection principles on one of its hyperspaces. We unify techniques and generalize theorems from the known results about selection principles for common hyperspace constructions. This includes results of Lj.D.R. Ko\v{c}inac, Z. Li, and others. We use selection games to generalize selection principles and we work with strategies of various strengths for these games. The selection games we work with are primarily abstract versions of the selection principles of Rothberger, Menger, and Hurewicz type, as well as games of countable fan tightness and selective separability. The hyperspace constructions that we work with are the Vietoris and Fell topologies, both upper and full, generated by ideals of closed sets. Using a new technique we are able to extend straightforward connections between topological constructs to connections between selection games related to those constructs. This extension process works regardless of the length of the game, the kind of selection being performed, or the strength of the strategy being considered.