Two point-picking games derived from a property of function spaces

Abstract

We present a study of two versions of the point-picking game defined by Berner and Juhasz. Given a space X there are two rivals O and P who take turns playing on X. In the n-^th round Player O takes a non-empty open subset U_n of the space X and P responds by choosing a point x_n ∈ U_n. After w-many moves are completed, the family L = {U_n, x_n, n ∈ w} is called the play of the game. In the CD-game CD(X) Player P wins if the set S(L) = {x_n : n ∈ w} is closed and discrete. Otherwise O is the winner. In the CL-game CL(X,p), where the point p ∈ X is fixed, Player O wins if S(L) contains p in its closure. If p ∉ S(L), then P is declared to be the winner. We show that in spaces Cp (X) both CD-game and CL-game are equivalent to Gruenhage's W-game for Player O. If π_x (p, X) ≤ w, then Player O has a winning strategy in CL (X, p). The converse is not always true. However, if X is separable or compact of π-weight ≤ w₁, then existence of a winning strategy for O in CL (X, p) is equivalent to π_x (p, X) ≤ w.

Mathematics Subject Classification (2010): Primary: 54C35; Secondary: 54C05, 54G20.

Keywords: Selection, point-picking games, function spaces, W-game, winning strategy, CD-game, CL-game, π-character