Stationary and convergent strategies in Choquet games

François G. Dorais¹ ; Carl Mummert²

¹ Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109, U.S.A.
² Department of Mathematics Marshall University 1 John Marshall Drive Huntington, WV 25755, U.S.A.

Fundamenta Mathematicae, Tome 209 (2010) no. 1, pp. 59-79

Résumé

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 $X$.

Détail

DOI : 10.4064/fm209-1-5

Keywords: nonempty has winning strategy against empty choquet game space space said choquet space winning strategy allows nonempty consider entire finite history previous moves before making each move stationary strategy only permits nonempty consider previous move empty nonempty has stationary winning strategy every second countable choquet space generally nonempty has stationary winning strategy choquet space open finite basis study convergent strategies choquet game proving following results space continuous image complete metric space only nonempty has convergent winning strategy choquet game nbsp space continuous compact image metric space only metacompact nonempty has stationary convergent strategy choquet game nbsp space continuous compact image complete metric space only metacompact nonempty has stationary convergent winning strategy choquet game nbsp