In this paper we introduce the notion of an F-metric, as a function valued distance mapping, on a set X and we investigate the theory of F-metric spaces. We show that every metric space may be viewed as an F-metric space and every F-metric space (X, ) can be regarded as a topological space (X, ). In addition, we prove that the category of the so-called extended Fmetric spaces properly contains the category of metric spaces. We also introduce the concept of an F¯-metric space as a completion of an F-metric space and, as an application to topology, we prove that each normal topological space is F¯-metrizable.

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