Stationary and convergent strategies in Choquet games.

*(English)*Zbl 1200.91054The 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)].

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)].

Reviewer: Laszlo Zsilinszky (Pembroke)