Uniform space and its hyperspace

In this paper, we examine some topological properties of uniform spaces and their hyperspaces. We prove that a uniform space $(X,\mathscr{U})$ is uniformly precompact if and only if $ (\exp_{c}X, \exp_{c}\mathscr{U})$ is uniformly precompact. Also we prove that the uniform hyperspace $(\exp_{c}X, \exp_{c} \mathscr{U})$ preserves uniformly local compactness, uniform connection, uniform paracompactness, and uniform $R$-paracompactness.