### Intrinsic shape - the proximate approach

Nikita Shekutkovski
2015 Filomat
In this paper we present the intrinsic approach to shape based on proximate sequences and nets of functions, and establish equivalence of different definitions. At the end are presented the results obtained recently, by use of the intrinsic approach to shape. The notion of shape was introduced by K. Borsuk in 1968 as a new classifcation of spaces from the point of view of their most important global topological properties. Main references about shape are the books of Borsuk  and of Mardesic
more » ... nd Segal  The approaches in both books are using external elements for describing shape of a space: neighborhoods in some external space where the original space is embedded, or an inverse sequence (system) of ANRs or polyhedra. On the other hand there exists an internal characterization of shape, an approach without external spaces. In this intrinsic approach to shape there are two main branches: 1) approach by proximate sequences and proximate nets 2) approach by multivalued functions. About the second approach we refer to articles: , , ,  and  . The subject of this paper is the intrinsic approach by proximate sequences and nets, and the results obtained by this approach. More detailed explanation of the papers connected with this approach is presented in Sections 2 and 3. Intrinsic approach to shape by proximate sequences and nets The idea of ε -continuity (continuity up to ε > 0) leads to continuity up to some covering V i.e., Vcontinuity, and the corresponding V -homotopy. Here, will be presented a short description of the intrinsic approach presented in  and  . Let X, Y be topological spaces. For collections U and V of subsets of X, U ≺ V means that U refines V, i.e., each U ∈ U is contained in some V ∈ V. By a covering we understand a covering consisting of open sets. A function is V -continuous, if it is V -continuous at every point x ∈ X . In this case, the family of all U x form a covering of X . By this, f : X → Y is V -continuous if there exists a covering U of X, such that for any x ∈ X,