Some Topological Properties of a Functor of Finite Degree

In this paper, the connection between a finally compact, pceudocompact, extremely disconnected, $$\aleph$$ -space and its hyperspace is studied. The action of functors $$\exp_{n},\exp_{c},\exp$$ on finally compact, pceudocompact, extremely disconnected and $$\aleph$$ -spaces is investigated. Some topological properties of uniformly space and its hyperspace is studied. It is proved: if the uniform space $$(X,\mathcal{U})$$ is uniformly paracompact, then $$\left(\exp_{c}X,\exp_{c}\mathcal{U}\right)$$ is uniformly paracompact. It is also shown: if the uniform space $$(X,\mathcal{U})$$ is uniformly $$R$$ -paracompact, then a uniform space $$\left(\exp_{c}X,\exp_{c}\mathcal{U}\right)$$ is uniformly $$R$$ -paracompact.

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