The Baire property on the hyperspace of nontrivial convergent sequences
type: Item
instance of: scholarly article
title: The Baire property on the hyperspace of nontrivial convergent sequences
zbMATH Open document ID: 1480.54013
author: R. Rojas-Hernández
author: Salvador Garcia-Ferreira
author: Y. F. Ortiz-Castillo
published in: Topology and its Applications
publication date: 2021-09-27T00:00:00Z
review text: A space in this paper is assumed to be a Fréchet-Urysohn nondiscrete completely regular Hausdorff space. An infinite countable subset \(S\) of a space \(X\) is called a nontrivial convergent sequence if \(S\) has a unique non-isolated point \(x_S\) such that \(S \setminus U\) is finite for each neighborhood \(U\) of \(x_S\). Let \(\mathcal{S}_c(X)\) denote the hyperspace consisting of all nontrivial convergent sequences of a space \(X\) with the Vietoris topology. In this paper, the authors give a characterization of \(\mathcal{S}_c(X)\) being a Baire space and a characterization of \(\mathcal{S}_c(X)\) being pseudocompact, which answers questions posed in [the authors, Topol. Proc. 52, 265--279 (2018; Zbl 1393.54011)]. In fact, the authors define a topological game \(GS(X,\beta,\alpha)\) of two players \(\alpha\) and \(\beta\) like the Banach-Mazur game, and prove that \(\mathcal{S}_c(X)\) is a Baire space if and only if \(X\) has a dense set of isolated points and \(X\) does not admit a winning strategy for player \(\beta\) in the game \(GS(X,\beta,\alpha)\). As corollaries of this theorem, the following are obtained: if a space \(X\) is completely metrizable and has a dense subset of isolated points, then \(\mathcal{S}_c(X)\) is a Baire space; if \(X\) is a scattered metric space, then \(\mathcal{S}_c(X)\) is a Baire space. It is also proved that \(\mathcal{S}_c(X)\) is pseudocompact if and only if \(X\) is homeomorphic to the one-point compactification of some uncountable discrete space.
reviewed by: Takamitsu Yamauchi
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 54G12
zbMATH DE Number: 7401491
zbMATH Keywords: hyperspace
zbMATH Keywords: Vietoris topology
zbMATH Keywords: nontrivial convergent sequence
zbMATH Keywords: one-point compactification
zbMATH Keywords: completely metrizable space
zbMATH Keywords: Baire space
zbMATH Keywords: scattered space
zbMATH Keywords: pseudocompactness
zbMATH Keywords: weakly pseudocompactness
zbMATH Keywords: Banach-Mazur game
MaRDI profile type: Publication
full work available at URL: j.topol.2020.107505
OpenAlex ID: W3111916498
cites work: Q3928648
cites work: Path connectedness, local path connectedness and contractibility of $\mathcal{S}_c(X)$
cites work: The Baire property in hit-and-miss hypertopologies
cites work: Baire spaces and Vietoris hyperspaces
cites work: Baire spaces, Tychonoff powers and the Vietoris topology
cites work: Remarks on dyadic spaces, II
cites work: Q3994935
cites work: Every scattered space is subcompact
cites work: Q4297859
cites work: The hyperspace of convergent sequences
cites work: Categorical properties on the hyperspace of nontrivial convergent sequences
cites work: Connectedness like properties on the hyperspace of convergent sequences
cites work: Subsequential filters
cites work: Selections and hyperspaces
cites work: Q4113079
cites work: Q4863330
cites work: Induced mappings on the hyperspace of convergent sequences
cites work: Cardinal functions of the hyperspace of convergent sequences
cites work: Q4600952
cites work: Topologies on Spaces of Subsets
cites work: A Note on Completely Metrizable Spaces
cites work: Topological games: on the 50th anniversary of the Banach-Mazur game
cites work: Baire spaces and hyperspace topologies
cites work: Q3402537
cites work: Baire spaces and hyperspace topologies revisited
DOI: 10.1016/J.TOPOL.2020.107505
Recommended article: Q4600952
Recommended article: Connectedness like properties on the hyperspace of convergent sequences
Recommended article: Baire property of spaces of continuous functions
Recommended article: Categorical properties on the hyperspace of nontrivial convergent sequences
Recommended article: The hyperspace of convergent sequences
Recommended article: On (strong) α-favorability of the Vietoris hyperspace
Recommended article: On (strong) \(\alpha \)-favorability of the Wijsman hyperspace
Recommended article: Compactness properties defined by open-point games
Recommended article: Baire spaces, Tychonoff powers and the Vietoris topology
Recommended article: Q5884620
description: scientific article; zbMATH DE number 7401491