Stationary and convergent strategies in Choquet games
Tom 209 / 2010
Streszczenie
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 $T_1$ Choquet space. More generally, Nonempty has a stationary winning strategy for any $T_1$ Choquet space with an open-finite basis.
We also study convergent strategies for the Choquet game, proving the following results. A $T_1$ space $X$ is the open continuous image of a complete metric space if and only if Nonempty has a convergent winning strategy in the Choquet game on $X$. A $T_1$ space $X$ is the open continuous compact 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 $T_1$ space $X$ is the open continuous compact 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 $X$.
