For a space $X$, let $\exp_fX$, $\exp_{lf}X$, $\exp_{clf}X$ be the collection of all nonempty closed subsets of $X$ with the finite, locally finite, countable locally finite topology, respectively. Some separations properties of the space $\exp_{lf}X$ are investigated. If $\exp_fX$ is a real-compact space, then $\exp_{clf}X$ is a real-compact space. If $X$ is a Lindelof space, then $\exp_{lf}X$ is a real-compact space. A simple example show that there exists a non Lindelof space $X$ such that $\exp_{lf}X$ is a real-compact space.