Variations on a game of Gale (III): remainder strategies

Extract

An infinite set X is given. D. Gale, in correspondence with J. Mycielski, described the following game in which players one and two play an inning per positive integer: In the nth inning one chooses a finite subset X_n of X, and two chooses a point x_n from (X₁∪ … ∪X_n)\{x₁,…,x_n−1}. A play

is won by two if . Gale asked whether two could have a winning strategy which depends for each n on knowledge of only the contents of the set

In mathematical terms, is there a function F defined on the collection of finite subsets of X such that:

for every sequence X₁, x₁, …, X_n, x_n,…. where each X_n is a finite subset

of X and for each n

we have

We shall call a strategy of this sort a remainder strategy for two. If there is some finite subset U of X such that F(U) ∉ U, then F cannot be a winning remainder strategy for two, because one can defeat F by choosing U each inning. So, when studying remainder strategies for two we may as well assume that for each finite set U ⊂ X, F(U) ∈ U.