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 \(f_{|{\operatorname{epi}}X|} = \pi^2 \circ \varphi_X = \mathcal{E} f_X \circ \varphi_X\).