On open-open games of uncountable length

TL;DR

This paper studies the open-open game of uncountable length, introducing a new cardinal invariant nd analyzing its bounds and properties within certain classes of topological spaces.

Contribution

It introduces the cardinal number nd characterizes its bounds, also defining the class nd analyzing its stability under products.

Findings

ounds between nd the spread nd its successor.

Spaces in re closed under products.

or spaces in re bounded by ssuming certain conditions.

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 $X\in\mathcal C_\kappa$ .