Local connectivity functions on arcwise connected spaces and certain continua
type: Item
instance of: scholarly article
title: Local connectivity functions on arcwise connected spaces and certain continua
zbMATH Open document ID: 1102.54035
author: Sam B. jun. Nadler
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: This is a continuation of the author's paper [Topol. Proc. 28, 229--239 (2004; Zbl 1081.54030)]. In the paper under review, the author adds a statement (part (1) in the following theorem) to the main theorem in the cited paper: Theorem 7. For a continuum \(X\), the following three statements are equivalent: (1) every connected real-valued function on \(X\) is a local connectivity function; (2) every connected real-valued function on \(X\) is a connectivity function; (3) every connected real-valued function on \(X\) is a Darboux function; (4) \(X\) is a dendrite such that each arc in \(X\) contains only finitely many branch points of \(X\). The author introduces a nice concept called \(\varepsilon\)-splitting set, and uses it to prove the following characterizations: Theorem 13. Let \(X\) be an arc-like continuum. Then the following statements are equivalent: (1) every real-valued local connectivity function on \(X\) is a connected function; (2) every real-valued local connectivity function on \(X\) is a Darboux function; (3) every real-valued local connectivity function on \(X\) is a connectivity function; (4) \(X\) is an arc. Theorem 17. Let \(X\) be a circle-like continuum. Then the following statements are equivalent; (1) every real-valued local connectivity function on \(X\) is a Darboux function; (2) every real-valued local connectivity function on \(X\) is a connectivity function; (3) \(X\) is a simple closed curve. A key result of this paper, which is used in most of the results is: Lemma 1. Let \(X\) be a Peano continuum. Then every local connectivity function from \(X\) to a topological space \(Y\) is a connected function. With these results the author gives partial answers to questions of J. Stallings, posed in 1959.
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54C08
zbMATH DE Number: 5013540
zbMATH Keywords: arc-like (chainable) continuum
zbMATH Keywords: atriodic continuum
zbMATH Keywords: circle-like continuum
zbMATH Keywords: composant
zbMATH Keywords: connected function
zbMATH Keywords: Darboux function
zbMATH Keywords: dendrite
zbMATH Keywords: \(\varepsilon\)-splitting set
zbMATH Keywords: hereditarily arcwise connected
zbMATH Keywords: indecomposable
zbMATH Keywords: irreducible
zbMATH Keywords: monotone map
zbMATH Keywords: \(n\)-od
zbMATH Keywords: Peano continuum
zbMATH Keywords: terminal continuum
reviewed by: Sergio Macías
MaRDI profile type: Publication
full work available at URL: j.topol.2005.03.012
OpenAlex ID: W2018496833
cites work: Snake-like continua
cites work: Darboux continuity
cites work: Chainable continua and indecomposability
cites work: Equivalence of Connectivity Maps and Peripherally Continuous Transformations
cites work: Q3304890
cites work: Multicoherence Techniques Applied to Inverse Limits
cites work: Continua which a one-to-one continuous image of [0,∞)
cites work: Q4004123
cites work: Q5692784
cites work: Almost continuity
cites work: Fixed point theorems for connectivity maps
cites work: Q5517570
cites work: Monotone Decompositions of Continua Not Separated by Any Subcontinua
DOI: 10.1016/J.TOPOL.2005.03.012
description: scientific article; zbMATH DE number 5013540