The category PPRS(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 of PPRS(Δ). The embeddings yield a description of metric
spces and their most natural generalizations entirely in terms of convergence criteria.