Let $X,Y$ - compact metric spaces. $f: X \mapsto Y$ $-$ 1st class Borel mapping. Let $G: 2^Y \mapsto 2^X$, $G(A)=cl(f^{-1}(A)), A\in 2^Y$.
Q: Is $A \mapsto X \times A$ continuous mapping?
Q: The distance between point and set in Hausdorff metric is a continuous function, dont it?
Let $t\in G(A)$, $(t,A) \in M \subset X \times 2^Y$. Let $L_A= X \times {A} \subset X\times 2^Y$. Then $G(A)= pr_1 (L_A \cap M)$. If $M-$ $F_\sigma$ we can use property $M=\cup M_n$, where $M_n-$ closed, and get $L_A \cap \cup M_n$.
Q:How to show that $M$ is $F_\sigma$?