A $\Delta$-complex is a topological space $X$ with a family of maps

for each $n \ge 0$ such that ($\Delta^n$ is the standard $n$-simplex)

1. Each interior restriction of map $\sigma^n_\alpha$ is injective
2. Each point $x \in X$ lies in the image of interior restriction of $\sigma^n_\alpha$ for exactly one $\sigma^n_\alpha$
3. Each $i$-boundary restriction of $\sigma^n_\alpha$ is equal to some $\sigma^{n-1}_\beta$ where we identify the $i$-boundary with a $\Delta^{n-1}$ by the unique linear homeomorphism that preserves the order of the vertices
4. A subspace $A \subseteq X$ is open if and only if $(\sigma^n_\alpha)^{-1}(A) \subseteq \Delta^n$ is open for all $\sigma^n_\alpha$