Topological properties defined by games and their applications
type: Item
instance of: scholarly article
title: Topological properties defined by games and their applications
zbMATH Open document ID: 1010.54018
DOI: 10.1016/S0166-8641(01)00168-7
author: Jiling Cao
author: Warren B. Moors
author: I. L. Reilly
published in: Topology and its Applications
publication date: 2002-09-17T00:00:00Z
review text: The authors investigate two-person games, called \({\mathcal G}({\mathcal F})\)-games, concerning a filter (filterbase) \({\mathcal F}\) in a space \(X\), and study topological properties defined by them. For a filter (filterbase) \({\mathcal F}\) in a space \(X\), the \({\mathcal G}({\mathcal F})\)-game played in \(X\) between players \(A\) and \(B\) is defined as follows: First, player \(A\) chooses a point \(x_1\in X\), and player \(B\) must then respond by choosing a member \(F_1\in{\mathcal F}\). Next, player \(A\) must select another (possibly the same) point \(x_2\in F_1\) and in turn player \(B\) must again respond to this by choosing a member \(F_2\in{\mathcal F}\). Repeating this infinitely, \(A\) and \(B\) produce a sequence \(\{(x_n,F_n):n\in \mathbb{N}\}\) with \(x_{n+1}\in F_n\) for each \(n\in \mathbb{N}\), called a play of the \({\mathcal G}({\mathcal F})\)-game. Then, \(B\) is said to have won this play if the sequence \(\{x_n\}\) has a cluster point in \(X\), and otherwise, \(A\) is said to have won this play. A pair \(({\mathcal F},\sigma)\) of a proper filter (filterbase) \({\mathcal F}\) in a space \(X\) and a winning strategy \(\sigma\) in the \({\mathcal G}({\mathcal F})\)-game is called a \(\sigma\)-filter (\(\sigma\)-filterbase). A space \(X\) is said to have property \((\ast\ast)\) if \(\bigcap\{\overline{F}:F\in{\mathcal F}\}\neq\emptyset\) for each \(\sigma\)-filterbase \(({\mathcal F},\sigma)\) in \(X\). The main results are: (i) If \(({\mathcal F},\sigma)\) is a \(\sigma\)-filterbase in a regular space with property \((\ast\ast)\), then the set \(K=\bigcap\{\overline{F}:F\in{\mathcal F}\}\) is compact and every open set containing \(K\) contains some member of \({\mathcal F}\). (ii) Property \((\ast\ast)\) is productive. (iii) Every Dieudonné-complete space has property \((\ast\ast)\). (iv) The space of real-valued continuous functions on a compact Hausdorff space with the pointwise convergence topology has property \((\ast\ast)\). (i) …
reviewed by: Haruto Ohta
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54E20
zbMATH DE Number: 1801749
zbMATH Keywords: \({\mathcal G}({\mathcal F})\)-game
zbMATH Keywords: property \((\ast\ast)\)
zbMATH Keywords: upper semicontinuous
zbMATH Keywords: active boundary
zbMATH Keywords: kernel
zbMATH Keywords: closed map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1801749