Forty Years of Fuzzy Metrics

Abstract

Kramosil and Michalek gave in 1975 a concept of fuzzy metric M on a set X which extends to the fuzzy setting the concept of probabilistic metric space introduced by K. Menger. After, George and Veeramani (Fuzzy Sets Syst 64: 395–399, 1994) modified the previous concept and gave a new definition of fuzzy metric. In both cases the fuzzy metric M induces a topology \(\tau _M\) on X which is metrizable. In this paper we survey some results relative to both concepts. In particular, we focus our attention in the completion of fuzzy metrics in the sense of George and Veeramani, since there is a significative difference with respect to the classical metric theory (in fact, there are fuzzy metric spaces, in this sense, which are not completable), and also in fixed point theory in both senses because it is a high activity area.