A point-picking game

Abstract

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 inCG(X). We will say that PlayerPwins w if {xn:n∈N} is closed in X, otherwise, PlayerOwins 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).