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?