  It is shown that the characteristic classes of foliations that were defined
by Losik and that take values in the de~Rham cohomology of the space of
infinite order frames over the leaf space may be mapped to the characteristic
classes with values in the \v{C}ech-de~Rham cohomology of the leaf space
studied in details by Crainic and Moerdijk. This map is in general
non-injective. All constructions are done using Losik's approach to Gelfand
formal geometry. A similar result is obtained for the exotic characteristic
classes as well as for the group actions of the diffeomorphisms. As
illustrating examples, foliations of codimension one are discussed.

Differential Geometry;Algebraic Topology