Stationary and convergent strategies in Choquet games
نویسندگان
چکیده
If Nonempty has a winning strategy against Empty in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows Nonempty to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits Nonempty to consider the previous move by Empty. We show that Nonempty has a stationary winning strategy for every second countable T1 Choquet space. More generally, Nonempty has a stationary winning strategy for any T1 Choquet space with an open-finite basis. We also study convergent strategies for the Choquet game, proving the following results. A T1 space X is the open image of a complete metric space if and only if Nonempty has a convergent winning strategy in the Choquet game on X. A T1 space X is the compact open image of a metric space if and only if X is metacompact and Nonempty has a stationary convergent strategy in the Choquet game on X. A T1 space X is the compact open image of a complete metric space if and only if X is metacompact and Nonempty has a stationary convergent winning strategy in the Choquet game on X. Mathematics subject classification: Primary 90D42, 54D20; Secondary 06A10, 06B35
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