This is an extension of Theorem \[thm:main\] to regions in ${\mathsf{Per}}$ of a particular form and is obtained by providing an explicit homotopy contracting the preimage of the region to the preimage of a single diagram that lies in the region. Theorem \[thmmaindyn\], described above, follows almost directly from Theorem \[thm:mainNdelta\], and is presented in Section \[secapplication\].

The proof of Theorem \[thm:main\] is extremely technical, involving an induction proof based on the number of persistence points or equivalently the number of local extrema pairs and requires substantial bookkeeping of subsets of ${\mathbb{R}}^N$.

We begin in Section \[sec:background\] with the necessary background information on sub-level set filtrations and persistence diagrams. We also recall the Nerve Theorem, which is the main tool that we use for our eventual proof of contractbility of the preimage set. The new content is a basic form of bookkeeping provided by convex polytopes covering the preimage set. By doing so we provide an easy proof that the preimage set $C(\operatorname{cv}(z)^i)$ is a finite union of polytopes (Proposition \[prop:Cpolytope\]).

Unfortunately, we are unable to prove contractibility using the representation presented in Section \[sec:background\]. Thus, in Section \[secindexing\] we introduce more complicated class of coverings of $C(\operatorname{cv}(z)^i)$. Fortunately, this 