# What is the name for a set endowed with a Lipschitz structure?

I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the Lipschitz structure (which is invariant under bi-Lipschitz transformations of metric spaces).

By a Lipschitz structure on a set $X$ we understand a maximal family $\mathcal L$ of bi-Lipschitz equivalent metrics on $X$.

A Lipschitz space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a Lipschitz structure $\mathcal L$ on $X$.

Is this terminology standard or used by some authors? If yes, could you provide a reference. If not, what is the alternative terminology for such kind of structure?

• I'm not sure there's a common name. I think the convenient language is the categorical language, which comes with considering the category of metric spaces and Lipschitz maps, and for which there's no need to consider maximal (or rather saturated) families of metrics. – YCor Jan 28 '18 at 1:10
• But in the case of the topological structure the notion of topology was invented and this turned to be more general than just a set with a maximal family of topologically equivalent metric. The same situation with the uniformity. – Taras Banakh Jan 28 '18 at 6:51
• If you are willing to restrict yourself to the case where $X$ is a manifold, this topic was discussed in another MO question, with references to work of Sullivan and Donaldson: mathoverflow.net/questions/146678/… – Lee Mosher Jan 28 '18 at 16:59
• Is there anything to stop us from just calling it a "Lipschitz space"? – Robert Furber Jan 28 '18 at 19:23
• Just a comment that the term "Lipschitz space" already has an established meaning as a Banach space of Lipschitz functions on a metric space. – Nik Weaver Jan 20 at 16:46

In the theory of quasiconformal and quasisymmetric mappings on metric spaces, the collection of metrics that are quasisymmetric to a given metric on a space is sometimes called a conformal gauge''. (See, e.g., Heinonen's Lectures on Analysis on Metric Spaces.) You could therefore try "Lipschitz gauge" or "bi-Lipschitz gauge", but to be honest I have never heard these terms used (and maybe they are already used for something else).