  We introduce a notion of a topologically flat locally convex module, which
extends the notion of a flat Banach module and which is well adapted to the
nonmetrizable setting (and especially to the setting of DF-modules). By using
this notion, we introduce topologically amenable locally convex algebras and we
show that a complete barrelled DF-algebra is topologically amenable if and only
if it is Johnson amenable, extending thereby Helemskii-Sheinberg's criterion
for Banach algebras. As an application, we completely characterize
topologically amenable K\"othe co-echelon algebras.
