Metric 1-spaces

TL;DR

This paper introduces metric 1-spaces, a generalized framework extending metric spaces, with a new axiomatization that preserves topological properties and connects to metric geometry examples.

Contribution

It presents a novel axiomatization of metric spaces and generalizes it to metric 1-spaces, maintaining topological features and providing new geometric examples.

Findings

New axiomatization of metric spaces

Introduction of metric 1-spaces

Examples from metric geometry

Abstract

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and equivalent, axiomatization of metric space is given which is then generalized from a fresh point of view. Naturally arising examples from metric geometry are presented.