# François G. Dorais

### Stationary and convergent strategies in Choquet games

##### By F. G. Dorais and C. Mummert
###### Fundamenta Mathematicae 209 (2010), no. 1, 59–79

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 $X$.