A continuous path $\gamma(t),t\in[0,1]$ in the space of ${\bf Z}$-affine structures with singularities on a given compact space $B$ is given by:

1. 1.

   a continuous path $B_{t}^{pre-sing}$ in the space of all compact subsets of $B$,

2. 2.

   a ${\bf Z}$-affine structure on $B\setminus B_{t}^{pre-sing}$ for all $t\in[0,1]$

Notice that for each $t_{0}\in(0,1)$ and $x_{0}\in B\setminus B_{t_{0}}^{pre-sing}$ we can choose neighborhoods $U_{t_{0}}$ of $t_{0}$ and $U_{x_{0}}$ of $x_{0}$ such that $U_{x_{0}}\subset B\setminus B_{t}^{pre-sing}$ for all $t\in U_{t_{0}}$. Then we require that:

1. 3.

   if $U_{t_{0}}$ and $U_{x_{0}}$ are sufficiently small then the induced ${\bf Z}$-affine structure on $U_{x_{0}}$ does not depend on $t\in U_{t_{0}}$.