On Open-Open Games of Uncountable Length

Abstract EN

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 ?κ of topological spaces that can be represented as the inverse limit of κ-complete system {Xσ,πρσ,Σ} with w(Xσ)≤κ and skeletal bonding maps.

It is shown that product of spaces which belong to ?κ also belongs to this class and μ(X)≤κ whenever X∈?κ.