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 $\su(X)\leqμ(X)\leq\su(X)^+$. We also introduce the class ${\mathcal{C}}_{\kappa }$ of topological spaces that can be represented as the inverse limit of $\kappa$-complete system $\{X_{\sigma },{\pi }_{\rho }^{\sigma },\Sigma \}$ with $\w(X_σ)\leqκ$ and skeletal bonding maps. It is shown that product of spaces which belong to ${\mathcal{C}}_{\kappa }$ also belongs to this class and $\mu (X)\le \kappa$ whenever $X\in {\mathcal{C}}_{\kappa }$ .

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