Probabilistic convergence spaces and generalized metric spaces

The categoryPPRS(Δ), whose objects are probabilistic pretopological spaces which satisfy an axiom(Δ)and whose morphisms are continuous mappings, is introduced. Categories consisting of generalized metric spaces as objects and contraction mappings as morphisms are embedded as full subcategories ofPPRS(Δ). The embeddings yield a description of metric spces and their most natural generalizations entirely in terms of convergence criteria.