Topological and cardinal properties of covariant functors of finite degree

The report is devoted to the study of cardinal and topological properties of covariant functors of finite degree. It is proved that the functor ${{\exp }_{c}}$ preserves final compactness, extremal disconnection, and an example is constructed that the functor $\exp$ does not preserve final compactness. It is also proved that the uniform hyperspace $(\exp_c X, \exp_c \mathcal{U})$ preserves precompactness, uniform locally compactness, uniform connectivity, uniform $R$-paracompactness, uniform entanglement, and uniform $P$-precompactness.