A characterisation of probabilistic metrizability for approach spaces

TL;DR

This paper characterizes probabilistic metrizability in approach and uniform gauge spaces, providing new categorical descriptions and solving a previously open problem in the theory of probabilistic metric spaces.

Contribution

It offers the first characterization of probabilistic metrizability for approach spaces and develops an isomorphic categorical description of probabilistic metric spaces.

Findings

Characterization of probabilistic metrizable approach spaces

Categorical description of probabilistic metrizability for uniform gauge spaces

New isomorphic description of probabilistic metric spaces using sets with collections of distances

Abstract

Characterisations of metrizable topological spaces or metrizable uniform spaces are well known. A natural counterpart to being metrizable for topological spaces can be expressed in terms of probabilistic metrizability for approach spaces. The notion of a probabilistic metrizable approach space is based on a well known concrete functor $\Gamma$, as introduced in [9], from the category of probabilistic metric spaces with respect to a continuous arbitrary t-norm to the category of approach spaces. A characterization of those probabilistic metrizable approach spaces is still missing and in the first part of this paper we solve this problem. A natural counterpart to being metrizable for uniform spaces can be expressed in terms of probabilistic metrizability for uniform gauge spaces. In the second part of the paper we start from another concrete functor $\Lambda$, as described in [7], on the category of probabilistic metric spaces with respect to a continuous t-norm to the category of uniform gauge spaces. In a similar way as for the functor $\Gamma$ we obtain a characterisation of probabilistic metrizability of uniform gauge spaces. The last section of the paper is devoted to an isomorphic description of the category of probabilistic metric spaces. This problem is not new. Previous attempts in providing isomorphic descriptions of the category of probabilistic metric spaces worked with collections of (pseudo)metrics. These attempts were only formulated in restricted cases. Our isomorphic description is in terms of objects that are sets endowed with a collection of distances, where the distances involved do not satisfy the triangle inequality but fulfil a mixed triangle condition instead.