On open-open games of uncountable length

Abstract

The aim of this note is to investigate the open-open game of uncountable length. We introduce a cardinal number $μ(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\su(X)\leqμ(X)\leq\su(X)^+$. We also introduce the class $\mathcal C_κ$ of topological spaces that can be represented as the inverse limit of $κ$-complete system $\{X_σ,π^σ_ρ,Σ\}$ with $\w(X_σ)\leqκ$ and skeletal bonding maps. It is shown that product of spaces which belong to $\mathcal C_κ$ also belongs to this class and $μ(X)\leqκ$ whenever $X\in\mathcal C_κ$ .