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Stationary and convergent strategies in Choquet games

2010; Polish Academy of Sciences; Volume: 209; Issue: 1 Linguagem: Inglês

10.4064/fm209-1-5

1730-6329

François G. Dorais, Carl Mummert,

Rings, Modules, and Algebras

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. (1) 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. (2) 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. (3) 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.