On (strong) \(\alpha \)-favorability of the Wijsman hyperspace
type: Item
instance of: scholarly article
title: On (strong) \(\alpha \)-favorability of the Wijsman hyperspace
zbMATH Open document ID: 1207.54024
author: Leszek Piątkiewicz
author: Laszlo Zsilinszky
published in: Topology and its Applications
publication date: 2010-10-28T00:00:00Z
review text: Given a metric space \((X,d)\), let \(CL(X)\) denote the hyperspace of nonempty closed subsets of \(X\) endowed with the Wijsman topology. The authors investigate the Banach-Mazur and the strong Choquet game on \(CL(X)\) from the nonempty player's (i.e., \(\alpha \)'s or second player) perspective. Since the strategies for these games are related to some completeness properties of the spaces, through their study it is possible to explore completeness properties of the hyperspaces \(CL(X)\). In this paper, for the strong Choquet game it is shown that if \(X\) is a locally separable metrizable space, then \(\alpha \) has a (stationary) winning strategy on \(X\) iff it has a (stationary) winning strategy on \(CL(X)\) for each compatible metric on \(X\). The analogous result for the Banach-Mazur game does not hold, not even if \(X\) is separable, as it is shown that \(\alpha \) may have a (stationary) winning strategy on \(CL(X)\) for each compatible metric on \(X\), and may not have one on \(X\). It is also shown than there exists a separable 1st category metric space such that \(\alpha \) has a (stationary) winning strategy on \(CL(X)\). This answers Question 4.2 of \textit{J. Cao} and \textit{H. J. K. Junnila} [Proc. Am. Math. Soc. 138, No.~2, 769--776 (2010; Zbl 1192.54011)]. The paper ends with the following open problem: Does there exist a metrizable non-Baire space \(X\) such that \(CL(X)\) is a Baire space for each compatible metric on \(X\)?
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 5807471
zbMATH Keywords: Wijsman topology
zbMATH Keywords: ball topology
zbMATH Keywords: Baire space
zbMATH Keywords: Banach-Mazur game
zbMATH Keywords: strong Choquet game
zbMATH Keywords: (Strongly) \(\alpha \)-favorable space
zbMATH Keywords: Bernstein set
zbMATH Keywords: Baire metric
MaRDI profile type: Publication
full work available at URL: j.topol.2010.07.034
OpenAlex ID: W2013802687
cites work: Q4049797
cites work: Q4837693
cites work: A Polish Topology for the Closed Subsets of a Polish Space
cites work: Q4277838
cites work: Q5571891
cites work: Amsterdam properties of Wijsman hyperspaces
cites work: The Wijsman hyperspace of a metric hereditarily Baire space is Baire
cites work: Q5699247
cites work: Every Wijsman Topology Relative to a Polish Space is Polish
cites work: On the hyperspace of a non-separable metric space
cites work: Infima of hyperspace topologies
cites work: Stratégies gagnantes dans certains jeux topologiques
cites work: Convergence of Closed Subsets in a Topological Space
cites work: Q3994935
cites work: Stationary strategies in topological games
cites work: Equivalence among hypertopologies
cites work: Q4113079
cites work: Q3209840
cites work: Topology and descriptive set theory
cites work: Topologies on Spaces of Subsets
cites work: On (strong) α-favorability of the Vietoris hyperspace
cites work: Topological games: on the 50th anniversary of the Banach-Mazur game
cites work: Topological Spaces That are α-Favorable for a Player with Perfect Information
cites work: Convergence of Sequences of Convex Sets, Cones and Functions. II
cites work: Baire spaces and hyperspace topologies
cites work: Polishness of the Wijsman topology revisited
cites work: Topological games and hyperspace topologies
cites work: Q3402537
cites work: Q3647033
DOI: 10.1016/J.TOPOL.2010.07.034
Recommended article: On (strong) α-favorability of the Vietoris hyperspace
Recommended article: Q5699247
Recommended article: Polishness of the Wijsman topology revisited
Recommended article: On the hyperspace of a non-separable metric space
Recommended article: Usco sections and Choquet-completeness
Recommended article: Q4338512
Recommended article: Q3402537
Recommended article: Q4840121
Recommended article: Q4837693
Recommended article: Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space
description: scientific article; zbMATH DE number 5807471