On Principal Fuzzy Metric Spaces

Abstract

In this paper, we deal with the notion of fuzzy metric space (X,M,∗), or simply X, due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable and also that there exist p-Cauchy sequences which are not Cauchy. We prove that if every p-Cauchy sequence in X is Cauchy, then X is principal, and we observe that the converse is false, in general. Hence, we introduce and study a stronger concept than principal, called strongly principal. Moreover, X is called weak p-complete if every p-Cauchy sequence is p-convergent. We prove that if X is strongly principal (or weak p-complete principal), then the family of p-Cauchy sequences agrees with the family of Cauchy sequences. Among other results related to completeness, we prove that every strongly principal fuzzy metric space where M is strong with respect to an integral (positive) t-norm ∗ admits completion.

Fuzzy Metrics in Terms of Fuzzy Relations

“…Currently, there is growing interest in exploring the topological properties of fuzzy metrics, as this line of study holds promise not only for theoretical constructions but also for fixed-point theorems and various practical applications. Regarding the investigation of the topological properties of classical fuzzy metrics, extensive references can be found in [9][10][11][12][13][14][15][16][17][18][19][20][21]. Although fuzzy metrics have demonstrated successful applications in image processing problems [22][23][24], their full potential remains untapped.…”