The notion of fuzzy metric space (X,M,∗)
Valentín Gregori, Juan-José Miñana, Samuel Morillas and Almanzor Sapena.
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.