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.