A Note On The Completeness Of Probabilistic Quasi-metric Spaces

In 1973, Lawvere introduced the notion of Cauchy completeness for (enriched) categories and demonstrated that in the case of metric spaces (viewed as categories enriched over [0,∞]^op) this notion is equivalent to the usual completeness of metric spaces by Cauchy sequences. In this note it is shown that, when the triangle function is left-continuous, the Cauchy completeness of probabilistic quasi-metric spaces viewed as enriched categories is equivalent to the bicompleteness with respect to the symmetry topology. In the last section, the completability of a special kind of probabilistic quasi-metric spaces, the fuzzy metric space, is discussed. To this end, the concept of (?)-completion is introduced.