Some generalizations of metric spaces

In this paper we introduce a notion of projectively inductively closed functor (p.i.c.-functor).We give sufficient conditions for a functor to be a p.i.c.-functor.In particular, any finitary normal functor is a p.i.c.-functor.We prove that every preserving weight p.i.c.functor of a finite degree preserves the class of stratifiable spaces and the class of paracompact σ-spaces.The same is true (even if we omit a preservation of weight) for paracompact Σ-spaces and paracompact p-spaces.