Pedro Tirado - Instituto Universitario de Matematica Pura y Aplicada, Universidad Politecnica de Valencia, Camino de Vera s/n, ۴۶۰۲۲ Valencia, Spain

In [Fuzzy Sets and Systems ۲۷ (۱۹۸۸) ۳۸۵-۳۸۹], M. Grabiec in- troduced a notion of completeness for fuzzy metric spaces (in the sense of Kramosil and Michalek) that successfully used to obtain a fuzzy version of Ba- nachs contraction principle. According to the classical case, one can expect that a compact fuzzy metric space be complete in Grabiecs 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 Grabiecs 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 Grabiecs version of the Edelstein xed point theorem and dierent interesting facts on the topology of fuzzy metric spaces.

Fuzzy metric space, Cauchy sequence, G-completeness, Compactness, Fixed point theorem