Stationary and convergent strategies in Choquet games

TL;DR

This paper investigates stationary and convergent strategies in Choquet games, establishing conditions under which players have winning strategies in various topological spaces, linking these strategies to properties like metacompactness and open images of metric spaces.

Contribution

It proves that nonempty has stationary winning strategies in second countable T1 Choquet spaces and characterizes spaces with convergent strategies via open and compact images of metric spaces.

Findings

Nonempty has stationary winning strategies in second countable T1 Choquet spaces.

Spaces with convergent strategies are characterized as open images of complete metric spaces.

Metacompact T1 spaces with stationary convergent strategies are exactly the compact open images of metric spaces.

Abstract

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.