On strategies for selection games related to countable dimension

TL;DR

This paper investigates strategies in specific topological selection games to better understand countable and zero-dimensional spaces, generalizing known equivalences and addressing non-separable, non-metrizable cases.

Contribution

It analyzes perfect- and limited-information strategies in selection games and generalizes Telgársky's proof to relate zero-dimensional and finite-dimensional spaces.

Findings

Characterization of countable dimension via selection games

Generalization of game equivalence for zero-dimensional spaces

Insights into non-separable, non-metrizable space properties

Abstract

Two selection games from the literature, $G_c(\mathcal O,\mathcal O)$ and $G_1(\mathcal O_{zd},\mathcal O)$, are known to characterize countable dimension among certain spaces. This paper studies their perfect- and limited-information strategies, and investigates issues related to non-equivalent characterizations of zero-dimensionality for spaces that are not both separable and metrizable. To relate results on zero-dimensional and finite-dimensional spaces, a generalization of Telg\'{a}rsky's proof that the point-open and finite-open games are equivalent is demonstrated.