### Constructible Spaces over the Reals

• Definition. Let $$S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}$$ for some non-negative integer $$n$$ and let $$a_0 = -\infty$$ and $$a_{n+1} = \infty$$.
Then an $$S$$-skeleton $$X$$ for an $$\overline{{\mathbb{R}}}$$-space is given by the following data:

• For $$i = 1, \dots, n$$ a locally path-connected compact topological space $$V_i$$.

• For $$i = 0, \dots, n$$ a locally path-connected compact topological space $$E_i$$.

• For $$i = 1, \dots, n$$ two continuous maps $$l_i \colon E_i \rightarrow V_i$$ and $$r_{i-1} \colon E_{i-1} \rightarrow V_i$$.

For an $$S$$-skeleton $$X$$ as above we define the geometric realization $$|X|$$ to be $\left( \coprod_{i=1}^n V_i \times \{a_i\} \right) \coprod \coprod_{i=0}^{n} E_i \times [a_i, a_{i+1}] / \sim,$ where $$\sim$$ is the equivalence relation generated by $$(l_i(x), a_i) \sim (x, a_i)$$ for $$i = 1, \dots, n$$ and $$(r_i(x), a_{i+1}) \sim (x, a_{i+1})$$ for $$i = 0, \dots, n-1$$. Moreover we define $$f_X \colon |X| \rightarrow \overline{{\mathbb{R}}}$$ to be the map induced by the projection onto the second factor.

An $$\overline{{\mathbb{R}}}$$-space given by a continuous map $$g \colon Y \rightarrow \overline{{\mathbb{R}}}$$ is constructible if there is a finite subset $$S \subset {\mathbb{R}}$$ and an $$S$$-skeleton $$X$$, such that $$f_X \cong g$$ as $$\overline{{\mathbb{R}}}$$-spaces.

The $$S$$-skeleton $$X$$ is an $$S$$-skeleton for a bounded $${\mathbb{R}}$$-space if $$E_0 = \emptyset = E_n$$.

A bounded $${\mathbb{R}}$$-space $$g \colon Y \rightarrow {\mathbb{R}}$$ is constructible if there is a finite subset $$S \subset {\mathbb{R}}$$ and an $$S$$-skeleton $$X$$ with $$E_0 = \emptyset = E_n$$, such that $$f_X \cong g$$ as $$\overline{{\mathbb{R}}}$$-spaces.

1. Example. The following image depicts the geometric realization and the associated height function of an $$\{a_1, a_2, a_3\}$$-skeleton for an $$\overline{{\mathbb{R}}}$$-space.

Now let $$S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}$$ for some non-negative integer $$n$$ and let $$a_0 = -\infty$$ and $$a_{n+1} = \infty$$.

1. Definition. Let $$X$$ and $$X'$$ be two $$S$$-skeletons for an $$\overline{{\mathbb{R}}}$$-space and suppose we have the following data:

• For $$i = 1, \dots, n$$ a continuous map $$\varphi^v_i \colon V_i \rightarrow V'_i$$.

• For $$i = 0, \dots, n$$ a continuous map $$\varphi^e_i \colon E_i \rightarrow E'_i$$.

This data describes a homomorphism of $$S$$-skeletons for an $$\overline{{\mathbb{R}}}$$-space $$\varphi \colon X \rightarrow X'$$, if the two diagrams $\xymatrix{ V_i \ar[d]_{\varphi^v_i} & E_i \ar[d]^{\varphi^e_i} \ar[l]_{l_i} \\ V'_i & E'_i \ar[l]^{l'_i} }$ and $\xymatrix{ E_{i-1} \ar[r]^{r_{i-1}} \ar[d]_{\varphi^e_{i-1}} & V_i \ar[d]^{\varphi^v_i} \\ E'_{i-1} \ar[r]_{r'_{i-1}} & V'_i }$ commute for $$i = 1, \dots, n$$.

The composition of two homomorphisms of $$S$$-skeletons is defined by composing the individual maps $$\varphi^v_i$$ and $$\varphi^e_i$$.

In the first definition we described the geometric realization of an $$S$$-skeleton for an $$\overline{{\mathbb{R}}}$$-space. This picture is only complete, if we also describe a geometric pendant to any homomorphism between $$S$$-skeletons.

• Definition. For a homomorphism $$\varphi \colon X \rightarrow X'$$ of $$S$$-skeletons for $$\overline{{\mathbb{R}}}$$-spaces, we define the geometric realization $$|\varphi|$$ of $$\varphi$$ to be the map induced on the quotients defined by the maps:

• $$V_i \times \{a_i\} \rightarrow V'_i \times \{a_i\}, (p, a_i) \mapsto (\varphi^v_i (p), a_i)$$ for $$i = 1, \dots, n$$.

• $$E_i \times [a_i, a_{i+1}] \rightarrow E'_i \times [a_i, a_{i+1}], (p, t) \mapsto (\varphi^e_i(p), t)$$ for $$i = 0, \dots, n$$.

Altogether this defines a faithful functor $$|\_|$$ from the category of $$S$$-skeletons for $$\overline{{\mathbb{R}}}$$-spaces to the category of $$\overline{{\mathbb{R}}}$$-spaces.