Function valued metric spaces

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