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Nerve Theorem {#secnerve}
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The fundamental tool in our study of the topology of the persistence map preimage is the Nerve Theorem, below we recall it as Theorem \[thmnerve\]. It is elementary to observe that any preimage set of interest ($C(h)$) is a finite union of polytopes, see Proposition \[prop:Cpolytope\]. However, the polytopes constructing $C(h)$ can have arbitrary topology. By finding an appropriate family of sets that covering $C(h)$ (denoted in the sequel by $\Lambda_h$’s), we show that the nerve of that covering is the full simplex, and hence, $C(h)$ is homotopy equivalent to a point (contractible). See also Example \[exn5\]. Let $\|\cdot\|$ denote the support of a simplicial complex, let $[n]=\{1,2,\dots,n\}$. Let ${{\mathcal{A}}}= \{A_1, A_2,\dots, A_n\}$ be a family of sets. The *nerve* records the “intersection pattern” of ${{\mathcal{A}}}$. It is the simplicial complex denoted by ${{\mathcal{N}}}({{\mathcal{A}}})$, with vertex set $[n]$ and with simplices given by $${{\mathcal{N}}}({{\mathcal{A}}}) = \left\{ {{\mathcal{J}}}\subset[n]\colon\,\bigcap_{i\in{{\mathcal{J}}}}{A_i}\neq\emptyset\right\}.$$

\[thmnerve\] Let $K_1, K_2,\dots, K_n$ be subcomplexes of a finite simplicial complex $K$ that together cover $K$ (each simplex of $K$ is in at least one $K_i$), and let $A_i:=\|K_i\|$. Suppose that the intersection $\cap_{i\in {{\mathcal{J}}}}{A_i}$ is empty or contractible for 