Dimensions of ordered spaces and Lorentzian length spaces

TL;DR

This paper introduces a new dimension concept for ordered spaces that aligns with manifold dimensions, explores Lorentzian length spaces, and investigates related metric and collapse phenomena.

Contribution

It defines a novel dimension for ordered spaces, matching manifold dimensions, and establishes obstructions for injective monotone maps in Lorentzian length spaces.

Findings

Dushnik-Miller dimension of Minkowski spaces is countably infinite

A new dimension notion for ordered spaces matches manifold dimension

Existence of rushing Cauchy functions with specified zero loci

Abstract

After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the existence of injective monotonous maps between Lorentzian length spaces. Furthermore we induce metrics on Cauchy subsets, relate respective Hausdorff dimensions, prove existence of rushing Cauchy functions with a given Cauchy zero locus and consider collapse phenomena in this setting.