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**Spaces defined by topological games. II.**
*(English)*
Zbl 0558.54029

The author continues his study of the (two player) topological games G(K,X) introduced in ibid. 88, 193-223 (1975; Zbl 0311.54025) to which the reader is referred for a definition. Here K denotes a class of spaces which contains all singletons and all closed subsets of its members, while X is a topological space. All spaces of the article are assumed to be completely regular. The author associates certain topological properties with the existence of winning strategies for the game G(K,X) for player I or player II. The author proves four main types of results:

(1) The space X for which player I has a winning strategy for G(K,X) is the union of countably many K-scattered subsets, where a space Y is called K-scattered if for every nonempty closed \(E\subseteq Y\) there exists \(x\in E\) and an open neighborhood U of x in Y such that \(E\cap \bar U\in K.\)

(2) The author proves some reduction theorems concerning the actions of player I by introducing auxiliary games G#(K,X) and \(G+(K,X)\). An example of such a theorem is that player I has a winning strategy in \(G+(k,X)\) if and only if player I has a winning strategy in G(K,X).

(3) A covering characterization of spaces favorable for player II is also presented using the notion of K-cover, where an open collection \({\mathcal U}\) in a space X is called a K-cover provided that for each closed \(E\subseteq X\) with \(E\in K\) there is a \(U\in {\mathcal U}\) such that \(E\subseteq U\). Then the dual game \(G^*(K,X)\) of F. Galvin [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 26, 445-449 (1978; Zbl 0392.90101)] is shown to be equivalent to G’(K,X) and player II is shown to have a winning strategy in \(G^*(K,X)\) provided an indexed family \(\{U(t_ 1,...,t_ n):<t_ 1,...,t_ n>\in T^ n\) and \(n\in N\}\) of open sets exists and satisfies certain conditions involving K- covers.

(4) Let 1 denote the class of all singletons and let C denote the class of all finite spaces. Then the author shows that the games G(1,Y) and G(C,Y) are undetermined in ZFC. Here Y is a space due to R. Pol [Stud. Math. 64, 279-285 (1979; Zbl 0424.46011)]. In particular, he shows using the result of (1) above, that neither player I nor player II has a winning strategy in G(1,Y). Prior to this an undeterminacy result for G(1,X), X a subset of R, had been obtained using MA by Galvin (op. cit.).

(1) The space X for which player I has a winning strategy for G(K,X) is the union of countably many K-scattered subsets, where a space Y is called K-scattered if for every nonempty closed \(E\subseteq Y\) there exists \(x\in E\) and an open neighborhood U of x in Y such that \(E\cap \bar U\in K.\)

(2) The author proves some reduction theorems concerning the actions of player I by introducing auxiliary games G#(K,X) and \(G+(K,X)\). An example of such a theorem is that player I has a winning strategy in \(G+(k,X)\) if and only if player I has a winning strategy in G(K,X).

(3) A covering characterization of spaces favorable for player II is also presented using the notion of K-cover, where an open collection \({\mathcal U}\) in a space X is called a K-cover provided that for each closed \(E\subseteq X\) with \(E\in K\) there is a \(U\in {\mathcal U}\) such that \(E\subseteq U\). Then the dual game \(G^*(K,X)\) of F. Galvin [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 26, 445-449 (1978; Zbl 0392.90101)] is shown to be equivalent to G’(K,X) and player II is shown to have a winning strategy in \(G^*(K,X)\) provided an indexed family \(\{U(t_ 1,...,t_ n):<t_ 1,...,t_ n>\in T^ n\) and \(n\in N\}\) of open sets exists and satisfies certain conditions involving K- covers.

(4) Let 1 denote the class of all singletons and let C denote the class of all finite spaces. Then the author shows that the games G(1,Y) and G(C,Y) are undetermined in ZFC. Here Y is a space due to R. Pol [Stud. Math. 64, 279-285 (1979; Zbl 0424.46011)]. In particular, he shows using the result of (1) above, that neither player I nor player II has a winning strategy in G(1,Y). Prior to this an undeterminacy result for G(1,X), X a subset of R, had been obtained using MA by Galvin (op. cit.).

Reviewer: H.H.Wicke