  We completely elucidate the relationship between two invariants associated
with an ergodic probability measure-preserving (pmp) equivalence relation,
namely its cost and the minimal number of topological generators of its full
group. It follows that for any free pmp ergodic action of the free group on $n$
generators, the minimal number of topological generators for the full group of
the action is $n+1$, answering a question of Kechris.
