Reifenberg Theorem for Locally Finitely Almost Splitting Sets

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 biH\"older 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 biH\"older homeomorphism of the disk into the set of subsets of $\mathbb{R}^n$.