Short paths in homogeneous continua
type: Item
instance of: scholarly article
title: Short paths in homogeneous continua
zbMATH Open document ID: 0619.54027
DOI: 10.1016/0166-8641(87)90049-6
author: David P. Bellamy
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: A homogeneous continuum can be almost arcwise connected without being locally connected, e.g., the dyadic solenoid. There are many known examples of arcwise connected homogeneous continua and each of them is locally connected. K. Kuperberg has asked whether every such continuum must be locally connected. If so, then every such continuum must have all of the properties of locally connected continua. (Of course, being arcwise connected is one such property.) In this paper the author discovers another. If X is a homogeneous arcwise connected continuum and \(\epsilon\) is a positive number, then there exists a natural number n such that any two points of X can be joined in X by an \(\epsilon\)-chain of n subarcs. The author then raises the question: Does this imply that there exists a family P of paths in X such that (1) for x,y\(\in X\) some path in P joins x to y and (2) for every \(\epsilon >0\) there exists a natural number n such that each path in P is the union of the element of an \(\epsilon\)- chain of n elements? If so, is homogeneity necessary?
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54C99
zbMATH DE Number: 4005134
zbMATH Keywords: uniform path connectedness
zbMATH Keywords: arcwise connected homogeneous continua
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90049-6
OpenAlex ID: W2060708833
cites work: The Cyclic Connectivity of Homogeneous Arcwise Connected Continua
cites work: Transformation groups and \(C^ *\)-algebras
cites work: Q3920297
cites work: Q4101154
cites work: Q5541833
description: scientific article; zbMATH DE number 4005134