  We show that every AF-algebra is generated by a single operator. This was
previously unclear, since the invariant that assigns to a C*-algebra its
minimal number of generators lacks natural permanence properties. In
particular, it may increase when passing to ideals or inductive limits.
  To obtain a better behaved theory, we not only ask if a C*-algebra is
generated by $n$ elements, but also if generating $n$-tuples are dense. This
defines the generator rank, which we show has many natural permanence
properties: it does not increase when passing to ideals, quotients or inductive
limits.
