Functors and 1-soft mappings

The properties of the mappings F(f) :F(X)→F(Y), where f :X→Y is a 1-soft mapping of metric compact spaces and F a functor from a certain class of finitary covariant functors of finite degree containing expn and SPGn, are examined. A criterion for the existence of a neighborhood of a point in F(Y) such that the restriction of F(f) to the preimage of this neighborhood is 1-soft is given. The obtained results make it possible to find the degree of topological inhomogeneity of spaces F(X) for nonmetrizable AE(1)-compacta X homogeneous with respect to character.