# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Stationary and convergent strategies in Choquet games

### Tom 209 / 2010

Fundamenta Mathematicae 209 (2010), 59-79 MSC: Primary 90D42, 54D20; Secondary 06A10, 06B35. DOI: 10.4064/fm209-1-5

#### 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 $X$.

#### Autorzy

• François G. DoraisDepartment of Mathematics
University of Michigan
530 Church Street
Ann Arbor, MI 48109, U.S.A.
e-mail
• Carl MummertDepartment of Mathematics
Marshall University
1 John Marshall Drive
Huntington, WV 25755, U.S.A.
e-mail

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