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Stationary and convergent strategies in Choquet games. (English) Zbl 1200.91054
The Choquet game Ch($$X$$) is played on a topological space $$X$$ by two players, EMPTY and NONEMPTY, as follows: EMPTY starts the game, in round $$n<\omega$$, EMPTY chooses an open set $$U_n$$ and $$x_n\in U_n$$, while NONEMPTY chooses an open $$V_n$$ so that for $$n\geq 1$$, $$x_n\in V_n\subseteq U_n\subseteq V_{n-1}$$. Player EMPTY wins the run $$\langle x_n,U_n,V_n\rangle_{n<\omega}$$, provided $$\bigcap_{n<\omega} V_n=\emptyset$$, otherwise, NONEMPTY wins. Note that in the literature, the term strong Choquet game is also used for this game, while the term Choquet game is used for the so-called Banach-Mazur game. A winning strategy (stationary winning strategy, resp.) for NONEMPTY is a function $$\mathfrak S$$ that chooses an open set for NONEMPTY based on all the previous moves (based on just the previous move, resp.) of EMPTY, so that if NONEMPTY plays according to $$\mathfrak S$$, NONEMPTY wins in Ch($$X$$) regardless of EMPTY’s plays. It has been shown by G. Debs [Fundam. Math. 126, 93–105 (1985; Zbl 0587.54033)] that having a winning strategy for NONEMPTY does not always imply having a stationary winning strategy for NONEMPTY. On the other hand, it is known that the two notions coincide in a $$T_3$$ space with a base of countable order.
With a nice new argument in the main result, the authors demonstrate this coincidence in $$T_1$$ spaces with open-finite bases; in particular, in 2nd countable $$T_1$$ spaces, as well. The authors also study, and characterize, when NONEMPTY has convergent (stationary) strategies in Ch($$X$$) (i.e., (stationary) strategies $$\mathfrak S$$ such that if $$\langle x_n,U_n,V_n\rangle_{n<\omega}$$ is a run of Ch($$X$$) compatible with $$\mathfrak S$$, then $$\{V_n: n<\omega\}$$ forms a base of neighborhoods for the points of $$\bigcap_{n<\omega} V_n$$). The characterization of convergent winning strategies for NONEMPTY in Theorem 3.3 is known, see Theorem 6.4 in G. Debs and J. Saint Raymond [Mathematika 41, 117–132 (1994; Zbl 0854.46021)].

MSC:
 91A44 Games involving topology, set theory, or logic 54D70 Base properties of topological spaces 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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