Some Properties of the Covariant Functor Set of Exponential Type

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In 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 -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 TOPY 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.