[Paper Review] Closed Discrete Selection in the Compact Open Topology

Christopher Caruvana, Jared Holshouser|arXiv (Cornell University)|Dec 16, 2018

Advanced Topology and Set Theory6 references4 citations

TL;DR

This paper establishes an equivalence between the closed discrete selection game on $C_k(X)$, the space of continuous real-valued functions on $X$ with the compact-open topology, and a modified version of the compact-open game on $X$. The key contribution is that strategies in the closed discrete selection game on $C_k(X)$ correspond precisely to strategies in this modified game, linking selection principles on $C_k(X)$ to topological properties of $X$, its hyperspace of compact subsets, and related function spaces.

ABSTRACT

In 2017, Tkachuk isolated the closed discrete selection property while working on problems related to function spaces [15]. In this paper we will study the closed discrete selection property and the related games and strategies on $C_k(X)$. Clontz and Holshouser showed previously that the closed discrete selection game on $C_p(X)$ is equivalent to a modification of the point-open game on $X$. In this paper we show that the closed discrete selection game on $C_k(X)$ is equivalent to a modification of the compact-open game on $X$. We also connect discrete selection properties on $C_k(X)$ to a variety of other properties on $X$, $C_k(X)$, and hyperspaces of $X$.

Motivation & Objective

• To investigate the closed discrete selection property and its game-theoretic counterpart in $C_k(X)$, the space of continuous functions from $X$ to $\mathbb{R}$ with the compact-open topology.
• To establish a precise equivalence between strategies in the closed discrete selection game on $C_k(X)$ and strategies in a modified compact-open game on $X$.
• To connect discrete selection properties on $C_k(X)$ to topological properties of $X$, $C_k(X)$, and the hyperspace $\mathbb{K}(X)$ of compact subsets of $X$.
• To correct and extend prior results on $k$-covers and $\gamma_k$-covers in the context of selection principles and game equivalences.
• To explore the robustness of these equivalences through limited information and Markov strategies, and to identify open problems for future research.

Proposed method

• Adapts and extends techniques from Arens, Scheepers, and Kočinac on selection principles in function spaces.
• Applies a new duality framework for games developed by Clontz to relate the closed discrete selection game on $C_k(X)$ to a modified compact-open game on $X$.
• Imposes the convention that only non-trivial open covers are considered, and explicitly includes $\Lambda$-covers and $k$-covers in the analysis.
• Uses recursive construction of strategies and function sequences to demonstrate non-winning behavior in the game, proving implications between selection principles.
• Employs the Vietoris topology on $\mathbb{K}(X)$ to analyze compact sets and their covers, and uses basic open sets of the form $[U_0; U_1, \dots, U_n]$ to describe topological structure.
• Leverages homogeneity of $C_k(X)$ to relate strong countable fan tightness and strong countable dense fan tightness to selection principles.

Experimental results

Research questions

• RQ1 To what extent can the theory of closed discrete selection on $C_k(X)$ be generalized to $C_k(X, [0,1])$?
• RQ2 How can Propositions 43 and 45—concerning $k$-covers and $\gamma_k$-covers—be generalized beyond their current scope?
• RQ3 Does there exist a space $X$ where Player I has a winning strategy in $G_1(\mathscr{N}[K(X)], \neg\mathcal{K}_X)$ but not in the finite-open game or with pre-determined strategies?
• RQ4 To what degree do these equivalences and strategies extend to longer-length games beyond $\omega$-length plays?
• RQ5 What is the precise relationship between Markov strategies in $G_1(\mathscr{N}[K(X)], \neg\mathcal{K}_X)$ and winning strategies in the closed discrete selection game on $C_k(X)$?

Key findings

• The closed discrete selection game on $C_k(X)$ is equivalent to a non-trivial modification of the compact-open game on $X$, where only non-trivial covers are considered and $\Lambda$-covers are included.
• The selection principle $S_1(\mathcal{K}_X, \mathcal{K}_X)$ is equivalent to $S_1(\Omega_{C_k(X),\mathbf{0}}, \Omega_{C_k(X),\mathbf{0}})$, indicating that the game-theoretic and selection-theoretic properties align in $C_k(X)$.
• Strong countable fan tightness of $C_k(X)$ is equivalent to the non-existence of a winning strategy for Player I in $G_1(\mathscr{T}_{C_k(X)}, CD_{C_k(X)})$.
• The existence of a winning Markov strategy for Player II in $G_1(\mathscr{N}[K(X)], \neg\mathcal{K}_X)$ implies the existence of a winning Markov strategy in the closed discrete selection game on $C_k(X)$.
• The paper corrects errors in earlier results, including Theorem 47 and Propositions 43 and 45, and recovers the correct statements for $k$-covers and $\gamma_k$-covers.
• The equivalence between $S_1(\mathcal{D}_{C_k(X)}, \Omega_{C_k(X),\mathbf{0}})$ and the other selection principles confirms the robustness of the characterization across different game and selection settings.