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 $\mu(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\su(X)\leq\mu(X)\leq\su(X)^+$. We also introduce the class $\mathcal C_\kappa$ of topological spaces that can be represented as the inverse limit of $\kappa$-complete system $\{X_\sigma,\pi^\sigma_\rho,\Sigma\}$ with $\w(X_\sigma)\leq\kappa$ and skeletal bonding maps. It is shown that product of spaces which belong to $\mathcal C_\kappa$ also belongs to this class and $\mu(X)\leq\kappa$ whenever $X\in\mathcal C_\kappa$ .