Is there a characterization of convergence in Hausdorff metric in terms of that in the underlying metric?

Question

Let $(E, d)$ be a metric space and $\mathcal C$ the set of all non-empty compact subsets of $E$. We endow $\mathcal C$ with the Hausdorff metric $d_H$. It's well-known that the topology of $\mathcal C$ depends on that of $E$, not on $d$. One example to illustrate this independece is as follows. Let $A,A_n \in \mathcal C$ such that $A_n \to A$ in $d_H$. Then $$ \lim_n A_n = \{ \lim x_n \mid (x_n) \in (A_n)\text{ convergent}\}. $$

This means the limit point $A$ of the sequence $(A_n)$ is characterized only by the convergent sequences in $E$. My question is the following.

Is there a simple characterization of $d_H(A_n, A) \to 0$ in terms of convergent sequences in $E$?

Answer 1

There is indeed such a characterization. It is given in Section 17.6.3 of the book "Point Sets" by E. Cech. It says roughly that, when $E$ is compact, a necessary and sufficient condition for $d_H(A_n, A) \to 0$ is that $A$ coincides with the set of limit points $\{ \lim x_n \mid (x_n) \in (A_n)\text{ convergent}\}$. The exact statement is a bit more involved and uses a few definitions given in Section 8.8 of the same book.