\subseteq  {{\mathcal{S}_N}}(z,z_{j_2}) \subseteq \cdots \subseteq {{\mathcal{S}_N}}(z,z_{j_N}).$$ and denoted by ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$.

Since ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$ is a finite filtration of simplicial complexes we can use the classical results from persistent homology [@ZC; @comptop] to compute the persistent homology of ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$. Since for fixed $N$, ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$ is completely determined by $z$, we treat this as a map $${\mathsf{Dgm}}\colon {\mathbb{R}}^N\to {\mathsf{Per}}$$ where ${\mathsf{Per}}$ denotes the space of all persistence diagrams. Furthermore, there are a variety of topologies that can be put on ${\mathsf{Per}}$ such that ${\mathsf{Dgm}}$ becomes a continuous map [@chazal; @stability].

If we replace $\max{\left\{ {z_j,z_{j+1}}\right\}}$ by $\min{\left\{ {z_j,z_{j+1}}\right\}}$ in the definition of $f$ and set\
${{\mathcal{S}_N}}(z,\theta) := {\left\{ {\sigma\in{{\mathcal{S}_N}}: f(z,\sigma)\geq \theta}\right\}}$, then the appropriate modification of Definition \[defn:sublevelset\] leads to a superlevel-set filtration of ${{\mathcal{S}_N}}$. The results of this paper are equally applicable in this setting, but the proofs must be modified appropriately.

Since ${\mathsf{Per}}$ is not a linear space, ${\mathsf{Dgm}}$ is far from a linear function. However, some restrictions of its image can be obtained from the values of the coordinates of $z$. For this reason it is useful to think of $z$ as a real valued function defined over its coordinate indices, i.e. $z\colon {\left\{ {1,\ldots,N}\right\}}\to {\mathbb{R}}$ 