Dissimilarity, Quasimetric, Metric

Dissimilarity is a function that assigns to every pair of stimuli a nonnegative number vanishing if and only if two stimuli are identical, and that satisfies the following two conditions called the intrinsic uniform continuity and the chain property, respectively: it is uniformly continuous with respect to the uniformity it induces, and, given a set of stimulus chains (finite sequences of stimuli), the dissimilarity between their initial and terminal elements converges to zero if the chains' length (the sum of the dissimilarities between their successive elements) converges to zero. The four properties axiomatizing this notion are shown to be mutually independent. Any conventional, symmetric metric is a dissimilarity function. A quasimetric (satisfying all metric axioms except for symmetry) is a dissimilarity function if and only if it is symmetric in the small. It is proposed to reserve the term metric (not necessarily symmetric) for such quasimetrics. A real-valued binary function satisfies the chain property if and only if whenever its value is sufficiently small it majorates some quasimetric and converges to zero whenever this quasimetric does. The function is a dissimilarity function if, in addition, this quasimetric is a metric with respect to which the function is uniformly continuous.