Article

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

Published in: Quaestiones Mathematicae
Volume 41 , issue 6 , 2018 , pages: 729–743

DOI: 10.2989/16073606.2017.1399172

Author(s): Vladimir V. Tkachuk Departamento de Matemáticas,

Keywords: Primary: 54C35 , Secondary: 54C05 , 54G20 , Primary: 54C35 , Secondary: 54C05 , 54G20

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 ω-many moves are completed, the family is called the play of the game. In the CD-game CD(X) Player P wins if the set 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 contains p in its closure. If , then P is declared to be the winner. We show that in spaces C_p(X) both CD-game and CL-game are equivalent to Gruenhage’s W-game for Player O. If , 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 ≤ ω₁, then existence of a winning strategy for O in CL(X, p) is equivalent to .

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