Two examples concerning hyperspace retraction
type: Item
instance of: scholarly article
title: Two examples concerning hyperspace retraction
zbMATH Open document ID: 0652.54008
DOI: 10.1016/0166-8641(88)90058-2
author: Alejandro Illanes
published in: Topology and its Applications
publication date: 1988-01-01T00:00:00Z
review text: For any continuum X let C(X) be the hyperspace of subcontinua of X, with the Hausdorff metric. A function r: C(X)\(\to X\) is a selection on C(X) if and only if r is continuous and r(A)\(\in A\) for every \(A\in C(X)\). The author is interested in relationships between the following conditions on X: (1) X is a strong deformation retract of C(X), (2) X is a deformation retract of C(X), (3) X is a retract of C(X), (4) C(X) admits a selection. Evidently (1)\(\Rightarrow (2)\Rightarrow (3)\) and (4)\(\Rightarrow (3)\). Two examples constructed in the paper show that \((4)\nrightarrow (2)\) (thus also \((3)\nrightarrow (2)\) and \((4)\nrightarrow (1))\) and \((3)\nrightarrow (4)\).
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C15
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 4064017
zbMATH Keywords: dendroid
zbMATH Keywords: hyperspace of subcontinua
zbMATH Keywords: strong deformation retract
zbMATH Keywords: selection
reviewed by: Maria Moszyńska
MaRDI profile type: Publication
full work available at URL: 0166-8641(88)90058-2
OpenAlex ID: W1984144979
cites work: C(X) is not Necessarily a Retract of 2 x
cites work: A Nonlocally Connected Continuum X such that C(X) is a Retract of 2 x
cites work: Some retractions and deformation retractions on \(2^ X\) and C(X)
cites work: A Continuum X which is a Retract of C(X) but not of 2 X
cites work: Q3425662
cites work: Concerning Continuous Selections
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description: scientific article; zbMATH DE number 4064017