Space of closed subsets of locally compact metric space is locally compact

Question

Let $(X,d)$ be a metric space. Assume that $d$ is a bounded metric. Let $C(X)$ be the collection of closed subsets of $X$. The Hausdorff metric on $C(X)$ is defined by $$d_H(A,B)= \max \{\sup_{b \in B} d(A,b), \sup_{a \in A} d(a,B)\}.$$ where $d(A,b)= \inf_{a \in A} \{d(a,b) \}$.

It is well-known that $(C(X),d_H)$ is compact if $X$ is compact.

I wonder if $(C(X),d_H)$ is locally compact given $X$ is locally compact?

Could you give me some hints or reference for this question?

Answer 1

I think you could use Alexandroff extension. Your metrics would have to extend accordingly.