### A Skeleton for the Epigraph

Let $$S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}$$ for some non-negative integer $$n$$ and let $$X$$ be an $$S$$-skeleton for a bounded $${\mathbb{R}}$$-space.

• Definition. The epigraph $${\operatorname{epi}}X$$ of $$X$$ is the $$S$$-skeleton defined by

• $$\tilde{V}_i := f_X^{-1} ((-\infty, a_i])$$ for $$i = 1, \dots, n$$,

• $$\tilde{E}_0 = \emptyset$$, $$\tilde{E}_i := f_X^{-1} ((-\infty, a_i]) \coprod (E_i \times [a_i, a_{i+1}]) / (l_i(x), a_i) \sim (x, a_i)$$ for $$i = 1, \dots, n-1$$, and $$\tilde{E}_n := |X|$$, and

• $$\tilde{l}_i := \left[{\operatorname{id}}\coprod l_i \circ \pi^1\right]$$ and $$\tilde{r}_{i-1}$$ the canonical quotient map for $$i = 1, \dots, n$$.

• Example. The left graphic in the following picture shows an $$\{a_1, a_2, a_3\}$$-graph, which we view as an $$\{a_1, a_2, a_3\}$$-skeleton for a bounded $${\mathbb{R}}$$-space, by augmenting the sets of vertices and edges with the discrete topology.

We ignore the colors for now. The right graphic shows the corresponding epigraph.

Moreover there is a natural homomorphism from $$X$$ to $${\operatorname{epi}}X$$.

• Definition. We define the homomorphism $$\kappa_X \colon X \rightarrow {\operatorname{epi}}X$$ by the following data:

• $$\kappa^v_{X i} \colon V_i \rightarrow \tilde{V}_i, p \mapsto (p, a_i)$$ for $$i = 1, \dots, n$$.

• $$\kappa^e_{X i} \colon E_i \rightarrow \tilde{E}_i, p \mapsto (p, a_{i+1})$$ for $$i = 1, \dots, n-1$$.

• Example. In the previous example each vertex and edge of the $$\{a_1, a_2, a_3\}$$-graph is depicted in a different color. Now $$\kappa_X$$ maps each vertex or edge to the point of the same color in the epigraph.

Now we show that $$f_{{\operatorname{epi}}X}$$ and $$\mathcal{E} f_X$$ are naturally isomorphic.

• Definition. We define a continuous map $$\varphi_X$$ from $$| {\operatorname{epi}}X |$$ to the epigraph $${\operatorname{epi}}f_X$$ of the continuous map $$f_X$$. In some sense most of $$| {\operatorname{epi}}X |$$ can already be seen as a subset of $${\operatorname{epi}}f_X$$ and on this part we choose $$\varphi_X$$ to be the inclusion. The part where this does not work is $$\left( V_i \coprod (E_i \times [a_i, a_{i+1}]) / (l_i(x), a_i) \sim (x, a_i) \right) \times [a_i, a_{i+1}] / \sim$$ and on this part we define $$\varphi_X$$ to be $$(p, x, y) \mapsto \left(p, a_i + (x-a_i) \frac{y-a_i}{a_{i+1} - a_i}, y\right)$$.

1. Lemma. The map $$\varphi_X$$ is a homeomorphism and we have $$f_{|{\operatorname{epi}}X|} = \pi^2 \circ \varphi_X = \mathcal{E} f_X \circ \varphi_X$$.