  A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that
all the divisorial valuations centered in an analytically irreducible local
noetherian ring are linearly comparable to each other. In the present paper we
generalize this theorem to the case of Abhyankar valuations with archimedian
value semigroup. Indeed, we prove that in a certain sense linear equivalence of
topologies characterizes Abhyankar valuations with archimedian semigroups,
centered in analytically irreducible local noetherian rings. Then we show that
some of the classical results on equivalence of topologies in noetherian rings
can be strengthened to include linear equivalence of topologies. We also prove
a new comparison result between the Krull topology and the topology defined by
the symbolic powers of an arbitrary ideal.
