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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?

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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Jan 28, 2018 at 1:10
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    $\begingroup$ 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. $\endgroup$
    – Taras Banakh
    Commented Jan 28, 2018 at 6:51
  • $\begingroup$ 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/… $\endgroup$
    – Lee Mosher
    Commented Jan 28, 2018 at 16:59
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    $\begingroup$ Is there anything to stop us from just calling it a "Lipschitz space"? $\endgroup$
    – Robert Furber
    Commented Jan 28, 2018 at 19:23
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    $\begingroup$ Just a comment that the term "Lipschitz space" already has an established meaning as a Banach space of Lipschitz functions on a metric space. $\endgroup$
    – Nik Weaver
    Commented Jan 20, 2019 at 16:46

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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).

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These spaces were introduced by Whitehead and called "locally metric spaces" by Luukkainen and Väisälä. See the precise references in the answer to Objects whose morphisms are Lipschitz maps.

This is assuming that by "Lipschitz" you mean "locally Lipschitz", so the resulting spaces are not necessarily metric spaces, and there is no notion of "bounded subset".

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    $\begingroup$ The notion is very natural with Lipschitz, so there no reason to assume to believe that the OP meant something else. So I don't think this answers the question. $\endgroup$
    – YCor
    Commented Jan 20, 2019 at 13:39

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