Density and finiteness. A discrete approach to shape
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.
scientific article; zbMATH DE number 991206
scientific article