Space of continuous set-valued mappings with closed unbounded values

We consider a space of continuous multivalued mappings defined on a locally compact space ${\mathcal T}$ with countable base. Values of these mappings are closed not necessarily bounded sets from a metric space $(X,d(\cdot))$ in which closed balls are compact. The space $(X,d(\cdot))$ is locally compact and separable. Let $Y$ be a dense countable set from $X$. The distance $\rho(A,B)$ between sets $A$ and $B$ from the family $CL(X)$ of all nonempty closed subsets of $X$ is defined as
$$\rho(A,B)=\sum_{i=1}^\infty \frac{1}{2^i}\,\frac{\mid~d(y_i,A)-d(y_i,B)\mid}{1+\mid~d(y_i,A)-d(y_i,B)\mid},$$
where $d(y_i,A)$ is the distance from a point $y_i \in Y$ to the set $A$. This distance is independent of the choice of the set $Y$, and the function $\rho(A,B)$ is a metric on the space $CL(X)$. The convergence of a sequence of sets $A_n$, $n\ge 1$, from the metric space $(CL(X),\rho(\cdot))$ is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space $(CL(X),\rho (\cdot))$ and give necessary and sufficient conditions for the compactness of sets in this space. The space $C({\mathcal T}, CL(X))$ of all continuous mappings from ${\mathcal T}$ to $(CL(X),\rho (\cdot))$ is endowed with the topology of uniform convergence on compact sets from ${\mathcal T}$. We prove the completeness and separability of the space $C({\mathcal T}, CL(X))$ and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space $C(T,CCL(X))$, where $T=[0,1]$, $X$ is a finite-dimensional Euclidean space, and $CCL(X)$ is the space of all nonempty closed convex sets from $X$ with the metric $\rho(\cdot)$. This space plays a crucial role in the study of sweeping processes. A counterexample showing the significance of the assumption of the compactness of closed balls from $X$ is given.