Countable Fan Tightness and Selection Games in Group-Valued Function Spaces

TL;DR

This paper extends game-theoretic characterizations of covering properties from real-valued to group-valued function spaces, establishing equivalences between selection games on the space and tightness games on function spaces.

Contribution

It generalizes existing results from real-valued functions to arbitrary metrizable arc-connected groups, connecting selection principles with tightness properties in group-valued function spaces.

Findings

Player II has a winning strategy in the $oldsymbol{ ext{Omega-Menger}}$ game iff in the countable fan tightness game on $C_p(X,G)$

Equivalence between $oldsymbol{ ext{Omega-Rothberger}}$ game on $X$ and countable strong fan tightness game on $C_p(X,G)$

Game-theoretic tightness properties are independent of $G$, preserved under $G$-equivalence, and valid for Markov strategies.

Abstract

Game-theoretic characterizations of selection principles provide a powerful framework for analyzing covering properties through strategic interactions. For a Tychonoff space $X$ and a non-trivial metrizable arc-connected topological group $G$, we prove that Player~II has a winning strategy in the $\Omega$-Menger game on $X$ if and only if Player~II has a winning strategy in the countable fan tightness game on $C_p(X, G)$ at the identity function. The analogous equivalence is established between the $\Omega$-Rothberger game on $X$ and the countable strong fan tightness game on $C_p(X, G)$ at the identity function. These results extend the game-theoretic characterizations of Clontz from $G = \mathbb{R}$ to arbitrary metrizable arc-connected groups, and lift the selection-principle equivalences of Ko\v{c}inac to the game-theoretic setting. As consequences, we establish that the game-theoretic tightness properties of $C_p(X,G)$ are independent of $G$, preserved under $G$-equivalence, and remain valid for Markov strategies.