Stationary and convergent strategies in Choquet games
By F. G. Dorais and C. Mummert
Fundamenta Mathematicae 209 (2010), no. 1, 59–79
 mr: 2652592
 zbl: 1200.91054
 arxiv: 0907.4126
 doi: 10.4064/fm20915
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 openfinite basis.
We also study convergent strategies for the Choquet game, proving the following results.

A T1 space is the open image of a complete metric space if and only if NONEMPTY has a convergent winning strategy in the Choquet game on .

A T1 space is the compact open image of a metric space if and only if is metacompact and NONEMPTY has a stationary convergent strategy in the Choquet game on .

A T1 space is the compact open image of a complete metric space if and only if is metacompact and NONEMPTY has a stationary convergent winning strategy in the Choquet game on .