A domain-theoretic approach to fuzzy metric spaces

We introduce a partial order ⊑M⊑M on the set BXBX of formal balls of a fuzzy metric space (X,M,∧)(X,M,∧) in the sense of Kramosil and Michalek, and discuss some of its properties. We also characterize when the poset (BX,⊑M)(BX,⊑M) is a continuous domain by means of a new notion of fuzzy metric completeness introduced here. The well-known theorem of Edalat and Heckmann that a metric space is complete if and only if its poset of formal balls is a continuous domain, is deduced from our characterization.