Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 5 (2010), 321 -- 332


Madjid Mirzavaziri

Abstract. 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,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 `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.

2010 Mathematics Subject Classification: Primary 54E35; Secondary 54E70, 54A40, 46C05.
Keywords: Function valued metric; Positive element; Strictly positive element; F-completeness; F-metric space; Allowance set; F-Cauchy; F-completion; F-metrizable.

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Acknowledgement. This research was supported by a grant from Ferdowsi University of Mashhad; No. MP89156MIZ.


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Madjid Mirzavaziri
Department of Pure Mathematics, Ferdowsi University of Mashhad,
P.O. Box 1159--91775, Iran.
Centre of Excellence in Analysis on Algebraic Structures (CEAAS),
Ferdowsi University of Mashhad, Iran.