Δ-complex

Definition

For $n \ge 0$ let $\Delta^n$ be the standard $n$-simplex with ordered vertices $e_0, \dots, e_n$, and let $\mathring{\Delta}^n$ be its interior. For $0 \le i \le n$ with $n \ge 1$, let $\delta^i \colon \Delta^{n-1} \to \Delta^n$ be the affine map onto the $i$-th face that sends $e_0, \dots, e_{n-1}$ in order to the vertices $e_0, \dots, \widehat{e_i}, \dots, e_n$. Let $X$ be a topological space. A $\Delta$-complex structure on $X$ is a family of continuous maps $\sigma_\alpha \colon \Delta^{n_\alpha} \to X$ such that:

1. Each $\sigma_\alpha$ is injective on $\mathring{\Delta}^{n_\alpha}$, and each point of $X$ lies in $\sigma_\alpha(\mathring{\Delta}^{n_\alpha})$ for exactly one $\alpha$.
2. For each $\alpha$ with $n_\alpha \ge 1$ and each $i$, the map $\sigma_\alpha \circ \delta^i$ is $\sigma_\beta$ for some $\beta$ with $n_\beta = n_\alpha - 1$.
3. A subset $A \subseteq X$ is open if and only if $\sigma_\alpha^{-1}(A)$ is open in $\Delta^{n_\alpha}$ for every $\alpha$.

A $\Delta$-complex is a space with a $\Delta$-complex structure.