  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 $X\in\mathcal C_\kappa$ .
