Dimensions of ordered spaces and Lorentzian length spaces
Olaf M\"uller

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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals · Advanced Topology and Set Theory
