Uniform Convergence and Uniform Continuity in Generalized Metric Spaces

In the paper [9] we introduced the class of generalized metric spaces. These spaces simultaneously generalize ‘standard’ metric spaces, probabilistic metric spaces and fuzzy metric spaces. We show that every generalized metric space is, naturally, a uniform space. Thus we can use standard topological techniques to study, for example, probabilistic metric spaces. We illustrate this by proving a Fixed Point Theorem for uniform spaces, and interpret this result in the context of probabilistic metric spaces. A uniform structure (or uniformity) on a set X is a non empty set U of subsets of X ×X which satisfies the following axioms: