Some Properties of Final Structured Spaces

Abstract

The aim of the study was to exp;ore several properties of special kind of morphisms between structured spaces called identification mappings of structured spaces and prove some results related to them.  Let (M,τ,C )  be  a structured space in the sense of  Mostow  and let    f: (M,τ,C )  → N, where N is arbitrary, be a   function.  There is a unique differential structure on N determined by f called the final, or identification, differential structure, and the space N then called the final structured space; this structure is    greater than every differential structure  on    N    such  that    f   is  smooth. Methodology:  we provided mathematical proofs of several theorems related to final structured spaces. We investigate the composition   of two functions have final differential structures, the relation between the final structured space and its quotient space, and the bijective mappings of structured spaces. Study results to the following:  the composition of two identification mappings of structured spaces is also an identification mapping of structured spaces, a structured space N is a final structured space    whenever  N/R  is  a quotient  structured space , a bijection  f is  an  identification  mapping  of  structured spaces if and only if the mapping  f is  a  diffeomorphism.

Conclusion: The study has shown some properties of final structured spaces and quotient structured spaces. Moreover, the case when the identification mapping of structured spaces f is bijection is also investigated, it has been shown    that   the notions    of diffeomorphisms    and    identification    bijections   of    structured spaces are related.