Let $X$ be a Hausdorff space and $\mathcal Z(X)$ be the hyperspace of compact subsets of $X$. For a Hausdorff space $X$ let $kX$ be the space on the set $X$ generated by the family of $k$-closed subsets of the space $X$. In this paper we establish som of the properties of the spaces: $\mathcal Z(kX)$, $kZ(X)$, $\mathcal Z^{(n)}(kX)$ and so on. Among other, we prove that te following conditions are equivalent: $\mathcal Z(X)$ is a $k$-space. $\mathcal Z(n)(X)$ is a $k$-space. $\mathcal Z^{(\omega)}(X)$ is a $k$-space.