On the Topological and Uniform Structure of Diversities

Andrew Poelstra

doi:10.1155/2013/675057

Abstract

Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note, we consider the analytical properties of diversities, in particular the generalizations of uniform continuity, uniform convergence, Cauchy sequences, and completeness to diversities. We develop conformities, a diversity analogue of uniform spaces, which abstract these concepts in the metric case. We show that much of the theory of uniform spaces admits a natural analogue in this new structure; for example, conformities can be defined either axiomatically or in terms of uniformly continuous pseudodiversities. Just as diversities can be restricted to metrics, conformities can be restricted to uniformities. We find that these two notions of restriction, which are functors in the appropriate categories, are related by a natural transformation.

Highlights

The theory of metric spaces is well-understood and forms the basis of much of, modern analysis
In 1956, Aronszajn and Panitchpakdi developed the notion of hyperconvex metric spaces [1] in order to apply the Hahn-Banach theorem in a more general setting
Uniform spaces admit notions of uniform continuity, uniform convergence, and completeness which coincide with the standard notions when metric spaces are considered as uniform spaces

Summary

Introduction

The theory of metric spaces is well-understood and forms the basis of much of, modern analysis. Uniform spaces admit notions of uniform continuity, uniform convergence, and completeness which coincide with the standard notions when metric spaces are considered as uniform spaces This theory has been described in Bourbaki’s General Topology [7] as well as Kelley’s classic text [8]. We will describe uniform continuity, uniform convergence, Cauchy sequences, and completeness for diversities, and show that these can be characterized in terms of conformities, giving an abstract framework in which to analyze the uniform structure of diversities This is motivated by the observation that while diversities generalize metric spaces in a straightforward way ( they restrict to metric spaces), they can exhibit very nonsmooth behavior with respect to these spaces (cf Theorem 1). The existing tools for metric spaces are insufficient to get a handle on the behavior of diversities

Preliminaries

Comparison with Metrics

Completeness in Diversities and Metric Spaces

Conformities

Category Theory