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We study cross ratios from an axiomatic viewpoint, also known as the study ofM\"obius spaces. We characterise cross ratios induced by quasi-metrics in termsof topological properties of their image. Furthermore, we generalise thenotions of Cauchy-sequences and completeness to M\"obius spaces and prove theexistence of a unique completion under an extra assumption that, again, can beexpressed in terms of the image of the cross ratio.

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