On generalized homogeneity of curves of a constant order
type: Item
instance of: scholarly article
title: On generalized homogeneity of curves of a constant order
zbMATH Open document ID: 0851.54034
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-09-28T00:00:00Z
review text: A point \(x\) of a space \(X\) is said to have Menger-Urysohn order \(\kappa\) if \(\kappa\) is the smallest cardinal such that \(X\) has a local base at \(x\) of sets whose boundaries have cardinality \(\kappa\). If \(X\) has a local base at \(x\) of sets with finite boundary, with the supremum of the cardinalities of the boundaries being infinite, then the order of \(X\) at \(x\) is said to be \(\omega\). A space \(X\) is said to be homogeneous with respect to a class \(F\) of mappings if, for every \(x,y\in X\), there exists a continuous surjection \(f\in F\) such that \(f; X\to X\) and \(f(x) =y\). It is a result of Urysohn that if all points of a nondegenerate continuum are of the same order \(\kappa\), then \(\kappa\in \{2, \omega, \aleph_0, c=2^{\aleph_0}\}\). For \(\kappa=2\), such a continuum must be a simple closed curve. The author discusses the constructions of plane continua of constant order and classes of maps which either do not increase or do not decrease order. He presents several questions concerning the characterization of such continua and their generalized homogeneity properties. Among other results, it is shown that if a continuum \(X\) contains a point of order \(\omega\) and is homogeneous with respect to a class of mappings which either does not decrease or does not increase the order of points, then \(X\) is locally connected.
reviewed by: Wayne Lewis
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 645215
zbMATH Keywords: confluent map
zbMATH Keywords: open map
zbMATH Keywords: light map
zbMATH Keywords: Menger-Urysohn order
zbMATH Keywords: plane continua
zbMATH Keywords: generalized homogeneity
author: Janusz J. Charatonik
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072413
OpenAlex ID: W2002153317
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DOI: 10.1216/RMJM/1181072413
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description: scientific article; zbMATH DE number 645215