I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz"). The answer seems pretty clear, and proceeds like the definition of manifolds:
1) If $X$ is a topological space, a Lipschitz chart is a homeomorphism from an open subset of $X$ to a metric space. 2) Two Lipschitz charts are compatible if the corresponding two transition functions are Lipschitz. 3) A Lipschitz atlas on $X$ is a set of compatible Lipschitz charts whose domains cover $X$. 4) A Lipschitz space is a topological space equipped with a maximal Lipschitz atlas.
So my question is: what are these spaces called? (and why not "Lipschitz spaces"?). I'll be grateful for any reference.