Volume 11 - Volume 11
Some Properties of the Covariant Functor Set of Exponential Type
Abstract
n this paper, it is shown that the sets of all non-empty subsets Set( )X of a topological space X
with exponential topology is a covariant functor in the category of Top -topological spaces and
their continuous mappings into itself. It is shown that the functor Set is a covariant functor in the category of topological spaces and continuous mappings into itself, a pseudometric in the space
Set( )X is defined, and compact, connected, finite, and countable subspaces of Set( )X are distinguished. It also shows various kinds of connectivity, soft, locally soft, and n soft mappings in Set( )X . One interesting example is given for the Y TOP category. It is proved that the functor Set maps open mappings to open, contractible and locally contractible spaces and into contractible and locally contractible spaces. Next, we study the problem of the propagation of mappings in the space
Set( )X and distinguish which sets the basic open sets of the space Set( )X consist of. The following takes place: a) The Set functor is a covariant functor in the Top category; b) The functor Set :Top Top preserves the layers of a continuous mapping, that is, 1 1 (Set( )) ( ( )) Set(f f A f A ( )) .
c) The functor Set preserves the contractibility of topological spaces.
Paper Details
PaperID: 1743
Author's Name: Tursunbay Zhuraev, Alisher Umarov, Kamariddin Zhuvonov and Almira Latipova
Volume: Volume 11
Issues: Volume 11
Keywords: Functors, Mappings, Connection, Soft Mappings, Contractility, Homotopy, Absolute Extensors.
Year: 2021
Month: April
Pages: 1139-1152