# The length of some diagonalization games

Archive for Mathematical Logic 38 (2):103-122 (1999)

 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) Keywords Key words:infinite game, $\omega$–concentrated, point-open type, $\omega$–type, strong-fan type, density type Categories (categorize this paper) DOI 10.1007/s001530050117 Options Edit this record Mark as duplicate Export citation  Find it on Scholar Request removal from index Revision history

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## Citations of this work BETA

On Some Questions About Selective Separability.Liljana Babinkostova - 2009 - Mathematical Logic Quarterly 55 (5):539-541.

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