Glasnik Matematicki, Vol. 52, No. 2 (2017), 331-350.

INTRINSIC STRONG SHAPE FOR PARACOMPACTA

Beti Andonovic and Nikita Shekutkovski

Faculty of Technology and Metallurgy , Ss Cyril and Methodius University , 1000 Skopje, Macedonia
e-mail: beti@tmf.ukim.edu.mk

Faculty of Mathematics and Natural Sciences, Ss Cyril and Methodius University , 1000 Skopje, Macedonia
e-mail: nikita@pmf.ukim.mk


Abstract.   In this paper an intrinsic definition of strong shape for paracompact topological spaces is presented. At first a coherent proximate net f:X → Y is defined, indexed by finite subsets of normal coverings of Y, and then there is a homotopy between two coherent proximate nets defined. A definition of composition of classes of homotopies between two coherent proximate nets f : X → Y and g : Y → Z is given. Then it is proved that for any other choice of corresponding coverings, a function is obtained that is in the same class with the previously defined composition. The strong shape category is obtained, with paracompacta as objects, and the homotopy classes of coherent proximate nets as morphisms.

2010 Mathematics Subject Classification.   55P55, 54C56, 55Q07.

Key words and phrases.   Coherent proximate net, covering, homotopy class, strong shape.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.52.2.10


References:

  1. B. Andonoviḱ, N. Shekutkovski, Subcategory of metric compacta in the strong shape category, Mat. Bilten 34 (2010), 13-25.
    MathSciNet    

  2. K. Borsuk, Theory of shape, Polish Scientific Publishers, Warszawa, 1975.
    MathSciNet    

  3. F.W. Cathey, J. Segal, Strong shape theory and resolutions, Topology Appl. 15 (1983), 119-130.
    MathSciNet     CrossRef

  4. D.A. Edwards, H.M. Hastings, Cech and Steenrod homotopy theories with applications to geometric topology. Lecture Notes in Mathematics 542, Springer, Berlin Heidelberg New York, 1976.
    MathSciNet    

  5. J.E. Felt, ε-continuity and shape, Proc. Amer. Math. Soc. 46 (1974), 426-430.
    MathSciNet     CrossRef

  6. Yu. Lisica and S. Mardešić, Coherent prohomotopy and strong shape of metric compacta, Glas. Mat. Ser. III 20(40) (1985), 159-186.
    MathSciNet    

  7. S. Mardešić and J. Segal, Shape theory, North-Holland, Amsterdam, 1982.
    MathSciNet    

  8. S. Mardešić, Strong shape and homology, Springer-Verlag, Berlin, 2000.
    MathSciNet     CrossRef

  9. J.B. Quigley, An exact sequence from the n-th to the (n-1)-st fundamental group, Fund. Math. 77 (1973), 195-210.
    MathSciNet     CrossRef

  10. J.M.R. Sanjurjo, A noncontinuous description of the shape category of compacta, Quart. J. Math. Oxford Ser. (2) 40 (1989), 351-359.
    MathSciNet     CrossRef

  11. N. Shekutkovski, Intrinsic definition of strong shape for compact metric spaces, Topology Proc. 39 (2012), 27-39.
    MathSciNet    

  12. N. Shekutkovski, Shift and coherent shift in inverse systems, Topology Appl. 140 (2004), 111-130.
    MathSciNet     CrossRef

  13. N. Shekutkovski and B. Andonovik, Intrinsic definition of strong shape of strong paracompacta, Proceedings of IV Congress of the mathematicians of Republic of Macedonia, Struga, October 18-22, 2008, 2010, 287-299.

Glasnik Matematicki Home Page