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 continuous 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 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 T₁ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-5,
author = {Fran\c cois G. Dorais and Carl Mummert},
title = {Stationary and convergent strategies in Choquet games},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {59-79},
zbl = {1200.91054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-5}
}
François G. Dorais; Carl Mummert. Stationary and convergent strategies in Choquet games. Fundamenta Mathematicae, Tome 209 (2010) pp. 59-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-1-5/