Density and finiteness. A discrete approach to shape
type: Item
instance of: scholarly article
title: Density and finiteness. A discrete approach to shape
zbMATH Open document ID: 0870.54019
DOI: 10.1016/S0166-8641(97)00098-9
published in: Topology and its Applications
publication date: 1997-08-31T00:00:00Z
review text: The notion of shape was introduced by K. Borsuk in 1968 as a new classification of spaces from the point of view of their most important global topological properties. Borsuk's approach to this theory, as well as Morita's, are not intrinsic, in the sense that they require some external elements to introduce basic definitions. The problem of giving internal characterizations of shape has been attacked following two different lines. The first of them tried to give a noncontinuous description of the shape category (Felt, Sanjurjo, Kieboom, Čerin) and the second one aimed at a description based on the use of multivalued maps (Sanjurjo, Giraldo, Morón, Ruiz de Portal, Čerin). In this article the authors give a new intrinsic description of shape using single-valued continuous functions. They show that the notion of shape is intimately related to discrete maps defined on dense subsets of compacta. In fact, a complete description of the shape morphisms between compacta \(X\) and \(Y\) can be given by using the following two simple tools: (i) continuous maps \(f:D\to Y\), where \(D\) is an open dense subset of \(X\) and the image of \(f\) is finite; (ii) a number \(\rho(f, g)\) which measures the distance between two such maps \(f\) and \(g\). In order to define compositions and give a full description of the shape category a second coefficient \(\epsilon(f, g)\) is also introduced in the paper. Finally, a connection with the theory of dynamical systems is established where the authors prove that the concept of omega limit, a central notion in topological dynamics, is related to the notion of shape image introduced by Lisica.
reviewed by: Zvonko Čerin
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 55P55
zbMATH DE Number: 991206
zbMATH Keywords: discrete maps
zbMATH Keywords: shape morphisms
zbMATH Keywords: shape category
zbMATH Keywords: omega limit
zbMATH Keywords: topological dynamics
zbMATH Keywords: shape image
author: Antonio Giraldo
author: José Sanjurjo
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00098-9
OpenAlex ID: W2011463701
description: scientific article; zbMATH DE number 991206