For a metric space $X = (X,d)$ ,let $\mathrm{Cld}_H(X)$ be the space of all nonempty closed sets in $X$ with the topology induced by the Hausdorff extended metric: $$d_H(A,B) = \max\bigg\{\sup_{x\in B}d(x,A),\sup_{x\in A}d(x,B)\bigg\} \in [0,\infty].$$
On each component of $\mathrm{Cld}_H(X)$ , $d_H$ is a metric (i.e., $d_H(A,B) < \infty$ ). In this paper, we give a condition on $X$ such that each component of $\mathrm{Cld}_H(X)$ is a uniform AR (in the sense of E. Michael). For a totally bounded metric space $X$ , in order that $\mathrm{Cld}_H(X)$ is a uniform ANR,a necessary and sufficient condition is also given. Moreover, we discuss the subspace $\mathrm{Dis}_H(X)$ of $\mathrm{Cld}_H(X)$ consisting of all discrete sets in $X$ and give a condition on $X$ such that each component of $\mathrm{Dis}_H(X)$ is a uniform AR and $\mathrm{Dis}_H(X)$ is homotopy dense in $\mathrm{Cld}_H(X)$ .