  Given a space $X$ we investigate the descriptive complexity class $\G_X$ of
the set $\FF_0(X)$ of all its closed zero-dimensional subsets, viewed as a
subset of the hyperspace $\FF(X)$ of all closed subsets of $X$. We prove that
  $\max \{ \G_X; \ X \text{ analytic } \}=\pca $ and
  $\sup \{ \G_X; \ X \text{ Borel } \borm \xi\} \supseteq \Game \bora \xi$ for
any countable ordinal $\xi\geq1$. In particular we prove that there exists a
one-dimensional Polish subpace of $2^\wo\times \R^2$ for which $\FF_0(X)$ is
not in the smallest non trivial pointclass closed under complementation and the
Souslin operation $\mathcal A\,$.
