The length of some diagonalization games

Abstract.

For X a separable metric space and $\alpha$ an infinite ordinal, consider the following three games of length $\alpha$: In $G^{\alpha}_1$ ONE chooses in inning $\gamma$ an $\omega$–cover $O_{\gamma}$ of X; TWO responds with a $T_{\gamma}\in O_{\gamma}$. TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is an $\omega$–cover of X; ONE wins otherwise. In $G^{\alpha}_2$ ONE chooses in inning $\gamma$ a subset $O_{\gamma}$ of ${\sf C}_p(X)$ which has the zero function $\underline{0}$ in its closure, and TWO responds with a function $T_{\gamma}\in O_{\gamma}$. TWO wins if $\underline{0}$ is in the closure of $\{T_{\gamma}:\gamma<\alpha\}$; otherwise, ONE wins. In $G^{\alpha}_3$ ONE chooses in inning $\gamma$ a dense subset $O_{\gamma}$ of ${\sf C}_p(X)$, and TWO responds with a $T_{\gamma}\in O_{\gamma}$. TWO wins if $\{T_{\gamma}:\gamma<\alpha\}$ is dense in ${\sf C}_p(X)$; otherwise, ONE wins. After a brief survey we prove: 1. If $\alpha$ is minimal such that TWO has a winning strategy in $G^{\alpha}_1$, then $\alpha$ is additively indecomposable (Theorem 4) 2. For $\alpha$ countable and minimal such that TWO has a winning strategy in $G^{\alpha}_1$ on X, the following statements are equivalent (Theorem 9): a) TWO has a winning strategy in $G^{\alpha}_2$ on ${\sf C}_p(X)$. b) TWO has a winning strategy in $G^{\alpha}_3$ on ${\sf C}_p(X)$. 3. The Continuum Hypothesis implies that there is an uncountable set X of real numbers such that TWO has a winning strategy in $G^{\omega^2}_1$ on X (Theorem 10).