The uniform topology of rough Riemannian metrics

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Rough Riemannian structures have been identified in recent times as an important class of geometric invariances of analytic objects. Though some immediate facts are known, there has not been a systematic study of this set. The goal here would be to study this set by endowing it with an extended distance metric. This includes understanding limits of sequences in this space and relating this space to other well-known topologies, such as the measured-Gromov-Hausdorff topology.