  We describe the envelope C*-algebra associated to a partial action of a
countable discrete group on a locally compact space as a groupoid C*-algebra
(more precisely as a C*-algebra from an equivalence relation) and we use our
approach to show that, for a large class of partial actions of Z on the Cantor
set, the envelope C*-algebra is an AF-algebra. We also completely characterize
partial actions of a countable discrete group on a compact space such that the
envelope action acts in a Hausdorff space.
