the following proposition that is proven in Section \[sec:background\].

\[prop:singlePoints\] If $z\in{\mathbb{R}}^n$ satisfies assumption [**A**]{}, then each persistence point in ${\mathsf{Dgm}}(z)$ is unique.

As indicated above, given $A\subset {\mathsf{Per}}$ we are interested in the topology of ${\mathsf{Dgm}}^{-1}(A)$. For the moment we focus on the pre-image of a single persistence diagram for which it is useful to catalog the order of the local extrema.

The *critical value set* is given by $$\label{def:E+}
{{\mathcal{E}}}_+ := {\left\{ { h= (h_1,\ldots,h_n)\in {\mathbb{R}}^n : \text{$n$ is odd, and $h_{2k\pm 1} < h_{2k}$ for all $k$} }\right\}}.$$ Set ${{\mathcal{E}}}^n_+:= {\left\{ {h\in \mathcal{E}_+ : h\in{\mathbb{R}}^n}\right\}}$. We will also use the simpler notation ${{\mathcal{E}}}^n = {{\mathcal{E}}}^n_+$.

To obtain results concerning superlevel set persistence one requires the use of $$\label{def:E-}
{{\mathcal{E}}}_- := {\left\{ { h= (h_1,\ldots,h_n)\in {\mathbb{R}}^n : \text{$n$ is odd, and $h_{2k\pm 1} > h_{2k}$ for all $k$} }\right\}}$$

\[defcv\] We define a map $\operatorname{cv}\colon {\mathbb{R}}^N \to {{\mathcal{E}}}_+^n$ as follows. Assume $z\in {\mathbb{R}}^N$ has $n$ local extrema that are either interior extrema or boundary minima. Observe that this implies that $n$ is odd. Let ${i_j}$ be an index associated with the $j$-th extremum ordered such that $i_j < i_{j+1}$. Observe that $$\operatorname{cv}(z) := (z_{i_1},\ldots, z_{i_n})\in {{\mathcal{E}}}^n_+\subset{{\mathcal{E}}}_+.$$

\[ex:cv(z)\] Let 