  The classical Reifenberg's theorem says that a set which is sufficiently well
approximated by planes uniformly at all scales is a topological H\"older
manifold. Remarkably, this generalizes to metric spaces, where the
approximation by planes is replaced by the Gromov-Hausdorff distance. This fact
was shown by Cheeger and Colding in an appendix of one of their celebrated
works on Ricci limit spaces [8]. Given the recent interest around this
statement in the growing field of analysis in metric spaces, in this note we
provide a self contained and detailed proof of the Cheeger and Colding result.
Our presentation substantially expands the arguments in [8] and makes explicit
all the relevant estimates and constructions. As a byproduct we also shows a
biLipschitz version of this result which, even if folklore among experts, was
not present in the literature. This work is an extract from the doctoral
dissertation of the second author.
