A Topological Characterization of the Length of Paths

Abstract

Let T be a metrizable topological space. We speak of a universal functional of paths if with every path ß in T a (finite or infinite) number λ ß is associated. For any particular metrization of T the corresponding length of paths is an example of a non-negative universal functional. To different metrizations of T correspond, in general, different lengths. How are these lengths characterized among the non-negative universal functionals of paths? In other words, what properties of a functional λ are necessary and sufficient in order that there exist a metrization of T such that, for every path ß, the corresponding length is equal to λ ß? We widen the scope of the problem by admitting metrizations of T for which the distance is non-symmetric.