  A countably based profinite group can be naturally seen as a metric space
with respect to a given filtration, and thus, it has a well defined Hausdorff
dimension function. Barnea and Shalev found a group theoretical expression for
the Hausdorff dimension of the closed subgroups of a profinite group $G$,
opening, in this way, a bunch of possibilities to explore. In this paper we
generalize Barnea's and Shalev's result to arbitrary closed subsets of $G$.
