On compactness and G-completeness in fuzzy metric spaces

Abstract

[EN] In [Fuzzy Sets and Systems 27 (1988) 385-389], M. Grabiec introduced a notion of completeness for fuzzy metric spaces (in the sense of Kramosil and Michalek) that successfully used to obtain a fuzzy version of Banach´s contraction principle. According to the classical case, one can expect that a compact fuzzy metric space be complete in Grabiec´s sense. We show here that this is not the case, for which we present an example of a compact fuzzy metric space that is not complete in Grabiec´s sense. On the other hand, Grabiec used a notion of compactness to obtain a fuzzy version of Edelstein ´s contraction principle. We present here a generalized version of Grabiec´s version of the Edelstein fixed point theorem and different interesting facts on the topology of fuzzy metric spaces.