Hyperspace of the $\Pi$-complete spaces and maps

In the present paper we establish that the space $\exp_\beta X$ of compact subsets of a Tychonoff space $X$ is $\Pi$-complete if and only if $X$ is so. Further, we prove the Tychonoff map $\exp_\beta f: \exp_\beta X\to \exp_\beta Y$ is $\Pi$-complete if and only if a given map $f: X\to Y$ is $\Pi$-complete.