A point-picking game

In this article we study a version of the point-picking game defined by Berner and Juhász. Given a space X, the closed game CG(X) on X between Player O and Player P is played as follows: Player O chooses a non-empty open set U1 ⊂ X, then Player P chooses a point x1 ∈ U1, then Player O chooses a non-empty open set U2 ⊂ X, then Player P chooses a point x2 ∈ U2, and so on. An infinite sequence w = (U1, x1, U2, x2, ...) such that Un is a nonempty open set and xn ∈ Un for every n ∈ N is called a play in CG(X). We will say that Player P wins w if {xn : n ∈ N} is closed in X, otherwise, Player O wins w. We prove that if Player O does not have a Markov winning strategy in CG(X) then X is selectively closed. We show that if X is the σ-product {f ∈ {0, 1}ω1 : f−1(1) is finite }, then Player O has a winning strategy in CG(X), yet X is selectively closed and selectively discrete. We also construct a selectively discrete space with a stationary winning strategy for Player O in CG(X). ©Elsevier