Models for \(C_{p}(x)\) for atriodic continua
type: Item
instance of: scholarly article
title: Models for \(C_{p}(x)\) for atriodic continua
zbMATH Open document ID: 1112.54008
published in: Topology and its Applications
publication date: 2006-12-14T00:00:00Z
review text: A continuum is a compact connected metric space with more than two elements. Given a continuum \(X\), let \(C(X)\) be the hyperspace of subcontinua of \(X\), considered with the Hausdorff metric. Given a point \(p\in X\), let \(C_p(X)=\{ A\in C(X):p\in A\}\). A subcontinuum \(A\) of \(X\) is called a triod provided that there exists a subcontinuum \(B\) of \(A\) such that \(A-B\) has at least three components. The continuum \(X\) is atriodic provided that it contains no triods. Given a subset \(W\) of a hyperspace, it is usually useful to find a model for \(W\) (i.e., a topological space homeomorphic to \(W\) described in a simpler way). In this direction, \textit{P. Pellicer-Covarrubias} [Houston J. Math. 31, 403--426 (2005; Zbl 1085.54008)] proved that if \(X\) is an atriodic continuum and \(p\in X\), then there exists a continuous function \(g: [0,1]\to [0,1]\) such that \(g(0)=0=g(1)\) and \(C_p(X)\) is homeomorphic to the set \(L(g)=\{(r,s)\in [0,1]^2 : 0\leq s\leq g(r)\}\). She also asked if the converse holds. In the paper under review, the author answers Pellicer-Covarrubias' question in the positive by constructing, for each \(g\), an atriodic (in fact, chainable) continuum \(X\) and a point \(p\in X\) such that \(L(g)\) is homeomorphic to \(C_p(X)\).
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5080158
zbMATH Keywords: Arc-like
zbMATH Keywords: Atriodic
zbMATH Keywords: Containment Hyperspaces
zbMATH Keywords: Continuum
zbMATH Keywords: Hyperspaces
author: Jorge M. Martínez-Montejano
MaRDI profile type: Publication
full work available at URL: j.topol.2006.03.022
OpenAlex ID: W2043432647
cites work: A non-metric indecomposable continuum
cites work: Inverse limit spaces
cites work: Intervals of Continua which are Hilbert Cubes
cites work: Local connectedness of inverse limit spaces
cites work: Q4230677
cites work: Q5541833
cites work: Q3304890
cites work: Q3869981
cites work: Q4004123
cites work: Q4314779
cites work: Q5693036
DOI: 10.1016/J.TOPOL.2006.03.022
Recommended article: Q3593900
Recommended article: Q4454250
Recommended article: Q3162346
Recommended article: On a Characterization of W-Sets and the Dimension of Hyperspaces
Recommended article: Embedding suspensions into hyperspace suspensions
Recommended article: Atriodic homogeneous continua
Recommended article: Kelley continua and the hyperspace \({\lambda}(X)\)
Recommended article: TOTALLY CHAIN-TRANSITIVE ATTRACTORS OF GENERIC HOMEOMORPHISMS ARE PERSISTENT
Recommended article: On chainable continua with almost unique hyperspace
Recommended article: Q5693036
description: scientific article; zbMATH DE number 5080158