The length of some diagonalization games (Q1283129)

An \(\omega\)-cover \(\mathcal U\) of an infinite separable metric space \(X\) is an open cover of \(X\) (with \(X\notin \mathcal U\)) such that for every finite set \(F\subseteq X\) there is \(U\in \mathcal U\) with \(F\subseteq U\). For the infinite ordinal \(\alpha\), we define the 2-player game \(G^{\alpha}(X)\) on \(X\) as follows: players I and II play a move for each ordinal \(\gamma < \alpha\). At move \(\gamma\), player I chooses an \(\omega\)-cover \({\mathcal U}_{\gamma}\) of \(X\) and then player II chooses \(U_{\gamma}\in{\mathcal U}_{\gamma}\). Player II wins the game if \(\{U_{\gamma}:\gamma<\alpha\}\) is an \(\omega\)-cover of \(X\); otherwise player I wins. The \(\omega\)-type of \(X\), written \(\text{ tp}_{\omega}(X)\), is defined to be the least ordinal \(\alpha\) such that player II has a winning strategy in \(G^{\alpha}(X)\). The author proves a number of results concerning \(\text{ tp}_{\omega}(X)\): for instance, that \(\text{ tp}_{\omega}(X)\) is an additively indecomposable ordinal. There are other results relating \(\text{ tp}_{\omega}(X)\) to ordinals defined from similar games on \(X\) and on the space of continuous real-valued functions on \(X\) (with the topology of pointwise convergence). A number of examples are given, including (assuming the Continuum Hypothesis) one of a space \(X\) with \(\text{ tp}_{\omega}(X)={\omega}^2\).