{\mathbb{R}}^N$ is such that for $i\neq j$ it holds that

-   $C(\operatorname{cv}(z)^i)$, $C(\operatorname{cv}(z)^j)$ are not path connected, i.e. that for any $z\in C(\operatorname{cv}(z)^i)$ and $z'\in C(\operatorname{cv}(z)^j)$, there is no continuous path connecting $z$ with $z'$ in ${\mathsf{Dgm}}^{-1}({\mathsf{diag}})$.

-   each $C(\operatorname{cv}(z)^i)$ is characterized by the equivalence class of the associated chiral merge tree.

The proof follows naturally from the results in [@curry]. If two vectors in ${\mathsf{Dgm}}^{-1}({\mathsf{diag}})$ are such that the *chiral merge trees* associated with filtration functions $f(z,\cdot)$, $f(z',\cdot)$ are different. It follows that $z, z'$ have the same number of extrema, whose heights are determined by the persistence diagram ${\mathsf{diag}}$, but appearing in different permutations. Hance any continuous path modifying $z$ into $z'$ must change the heights of the local minima, and hence, the persistence diagram along this path must not be constant. Therefore, there is no continuous path in ${\mathsf{Dgm}}^{-1}({\mathsf{diag}})$, which connects $z$ with $z'$.

The case of super-level set filtrations is completely analogous, and we leave redoing the super-level case as an exercise for the reader.

Topology of $C(h)$ {#sec:C(h)}
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Let us fix example $h\in {{\mathcal{E}}}^n_+$. In this subsection we show that $C(h)$ defined in is the union of a finite set of polytopes.

The set 