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Möbius structures, quasi-metrics, and completeness

Merlin Incerti-Medici

Abstract

We study cross ratios from an axiomatic viewpoint, also known as the study of Möbius spaces. We characterise cross ratios induced by quasi-metrics in terms of topological properties of their image. Furthermore, we generalise the notions of Cauchy-sequences and completeness to Möbius spaces and prove the existence of a unique completion under an extra assumption that, again, can be expressed in terms of the image of the cross ratio.

Möbius structures, quasi-metrics, and completeness

Abstract

We study cross ratios from an axiomatic viewpoint, also known as the study of Möbius spaces. We characterise cross ratios induced by quasi-metrics in terms of topological properties of their image. Furthermore, we generalise the notions of Cauchy-sequences and completeness to Möbius spaces and prove the existence of a unique completion under an extra assumption that, again, can be expressed in terms of the image of the cross ratio.

Paper Structure

This paper contains 6 sections, 21 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Möbius structures and quasi-metrics
  4. The Möbius topology
  5. Cauchy-sequences and completeness
  6. Constructing the completion

Key Result

Theorem 1

Let $(Z, \rho)$ be a metric space, $M$ the Möbius structure induced by $\rho$. Denote the metric topology induced by $\rho$ by $\mathcal{T}_{\rho}$ and the Möbius topology induced by $M$ by $\mathcal{T}_{M}$. Then $\mathcal{T}_{\rho} = \mathcal{T}_{M}$.

Figures (2)

  • Figure 1: A Möbius structure $crt$ satisfies the (corner)-condition if and only if we can find open neighbourhoods as depicted above, such that the image of $crt$ in $\overline{\Delta}$ doesn't intersect these neighbourhoods.
  • Figure 2: A Möbius structure $crt$ satisfies the (symmetry)-condition if and only if no point in the boundary of $\overline{\Delta}$ can be approximated by a sequence of points in $\mathop{\mathrm{Im}}\nolimits(\Delta)$ except for $(\frac{1}{2} : \frac{1}{2} : 0), (\frac{1}{2} : 0 : \frac{1}{2})$ and $(0 : \frac{1}{2} : \frac{1}{2})$. In other words, the image doesn't touch the boundary at any other than those three points.

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 40 more