Abstract.We give a probabilistic generalization of the theory of generalized metric spaces [2]. Then, we prove a fixed point theorem for a self-mapping of a probabilistic generalized metric space, satisfying the very general nonlinear contraction condition without the assumption that the space is Hausdorff.

A necessary and sufficient condition for a probabilistic metric space to be complete is given and the uniform limit theorem [2] is generalized to probabilistic metric space.

By considering probabilistic quasi-(pseudo-)metric spaces, we introduce and study the concept of probabilistic Hausdorff quasi-(pseudo-)metric metric that extend the corresponding notions of probabilistic Hausdorff metric. Finally completeness property of this notion are explored.

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