M\"obius structures, quasi-metrics, and completeness

We study cross ratios from an axiomatic viewpoint, also known as the study of M"obius 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 M"obius 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.