Selective strong screenability and a game

TL;DR

This paper explores the differences between selective strong screenability and screenability games in metric spaces, identifying conditions for winning strategies and demonstrating their non-equivalence.

Contribution

It establishes that selective versions of these games are not equivalent in standard metric spaces and provides criteria for winning strategies.

Findings

Selective strong screenability and screenability games are not equivalent in the closed unit interval.

Conditions for ONE to have a winning strategy are identified.

Necessary conditions for TWO to have a winning strategy are established.

Abstract

Selective versions of screenability and of strong screenability coincide in a large class of spaces. We show that the corresponding games are not equivalent in even such standard metric spaces as the closed unit interval. We identify sufficient conditions for ONE to have a winning strategy, and necessary conditions for TWO to have a winning strategy in the selective strong screenability game