Shekutkovski, Nikita and Atanasova-Pacemska, Tatjana and Vasilevska, Violeta and Markoski, Gjorgi and Andonovikj, Beti and Misajleski, Zoran and Soptrajanov, Martin and Velkoska, Aneta (2012) Foundation of Shape Theory. [Project] (Submitted)
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Abstract
The main goal of the project is to lay foundations and develop the intrinsic definition of shape. The
theory of shape was founded about 40 years ago. Since that time more than a thousand research papers and
several monographs ([1] ,[2], [3], [4]) on shape theory were published all over the world. The two classical
approaches to shape theory are: 1) the original Borsuk's approach, based on embedding compact metric
spaces in the Hilbert cube and studying the fundamental sequences of maps ([1], [2]) and 2) the inverse
system approach ([3] and [4]). Both approaches use external spaces - neighbourhoods of spaces embedded
in the Hilbert cube, ANR spaces and polyhedra. The intrinsic definition of shape does not use external
spaces. The main tool in founding this third approach to shape theory are functions f : X →Y which are
near to continuous functions, i.e., functions which are continuous up to some coverings of the space Y .
The intrinsic definition of shape is defined using sequences or nets of such functions and cofinal
families of coverings, i.e., families such that for any covering of the space Y, in the family there exists a finer
element.
Introducing the corresponding notion of homotopy, the shape morphisms are homotopy classes of
sequences or nets of near continuous functions. The intrinsic definition of strong shape is introduced using
homotopies of higher order [5].
Using this new definition of strong shape we expect to prove some new theorems in the theory of
strong shape and solve some of the existing problems. Moreover, this new approach yields new proofs of
already existing theorems of the strong shape theory.
One of the main tasks of the proposed investigations will be to prove equivalence of the new definition
and the already existing definition of strong shape. Use of the intrinsic definition of proper shape should be
applicable to the shape of noncompact spaces, in particular, to locally compact metric spaces.
Moreover, some applications to topological dynamics are expected.
[1] K. Borsuk, THEORY OF SHAPE, Lecture Notes Series 28, Aarhus, 1971.
[2] K.Borsuk, THEORY OF SHAPE, Polish Scientific Publishers, 1975.
[3] S.Mardešić and J. Segal, SHAPE THEORY (The Inverse Sistem Approach), North-Holland Publishing
Company, 1982
[4] S.Mardešić, STRONG SHAPE AND HOMOLOGY, Springer, 2000.
[5] N. Shekutkovski, Intrinsic definition of strong shape for compact metrics spaces, Topology Proceedings,
39, 27-39
| Item Type: | Project |
|---|---|
| Subjects: | Natural sciences > Matematics |
| Divisions: | Faculty of Computer Science |
| Depositing User: | Tatjana A. Pacemska |
| Date Deposited: | 28 Jan 2013 15:29 |
| Last Modified: | 21 Oct 2014 13:23 |
| URI: | https://eprints.ugd.edu.mk/id/eprint/5064 |
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