Function valued metric spaces

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-metrics paces. 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.

Keywords

• Function valued metric
• Positive element
• Strictly positive element
• ℱ-completeness
• ℱ-metric space
• Allowance set
• ℱ-Cauchy
• ℱ-completion
• ℱ-metrizable