The well-known Reifenberg theorem states that if a subset of \(\mathbb {R}^n\) can be well approximated by k-planes at every point and every scale, then it is biHölder homeomorphic to a k-disk. This article concerns a subset S of \(\mathbb {R}^n\) which can be approximated by at most N parallel k planes at each point and scale. As a subset of \(\mathbb {R}^n\) such an S may be quite degenerate; S may clearly not be homeomorphic to a disk, and indeed we will see may not be homeomorphic to a union of disks. However, we prove that S is still the image of a multivalued map on \(\mathbb {R}^k\) , which is itself a biHölder homeomorphism of the disk into the set of subsets of \(\mathbb {R}^n\) .
