Suppose that $\Pi$ is a maximal dual $\Delta$-complex. A point $x$ in $\bar{\Pi}$ has a neighborhood of one of the following $\frac{(n+1)(n+2)}{2}$ types: $\mathbb{R}^{k}\times\Sigma^{l-k}\times[0,+\infty)^{n-l}$, where $k\leq l\leq n$. Here $k$ is the dimension of the open cell of $\bar{\Pi}$ which contains $x$ while $l+1$ is the dimension of the open face of $\Delta$ which contains $x$.