On Open-Open Games of Uncountable Length - Andrzej Kucharski (International Journal of Mathematics and Mathematical Sciences)
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)≤𝜇(𝑋)≤𝑐(𝑋)+. We also introduce the class 𝒞𝜅 of topological spaces that can be represented as the inverse limit of 𝜅-complete system {𝑋𝜎,𝜋𝜎𝜌,Σ} with 𝑤(𝑋𝜎)≤𝜅 and skeletal bonding maps. It is shown that product of spaces which belong to 𝒞𝜅 also belongs to this class and 𝜇(𝑋)≤𝜅 whenever 𝑋∈𝒞𝜅.