On open-open games of uncountable length

Abstract

The aim of this paper 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 c(X) ≤ (X) ≤ c(X) +. We also introduce the class C k of topological spaces that can be represented as the inverse limit of k-complete system {X σ, π ρ σ,Σ} with w (X σ) ≤ k and skeletal bonding maps. It is shown that product of spaces which belong to C k also belongs to this class and μ(X) ≤ k whenever X ∈ C k.