Selection Games on Continuous Functions

TL;DR

This paper investigates selection principles and game-theoretic properties of spaces of continuous functions under set-open topologies, unifying various techniques and extending previous results in the field.

Contribution

It introduces a unifying framework connecting selection principles, game duality, and topological concepts for continuous function spaces.

Findings

Established equivalences between selection principles and game properties.

Developed a unifying framework linking topological notions with game-theoretic concepts.

Extended techniques involving point-picking and topological duality to new settings.

Abstract

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [13] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on \(X\) and \(C_p(X)\), the current authors showed similar equivalences in [1] involving the compact subsets of \(X\) and \(C_k(X)\). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game.