Let $\mathcal{F}=(X,f,B)$ be a topological $T^{n}$ fibration with discriminant locus $\Delta\subseteq B$ and fibre $X_{b}$ over $b\in B$. We assume that $\mathcal{F}$ satisfies the following conditions:

1. 1.

   for $n=2$, $\Delta$ is a finite union of points and given a small neighborhood $U$ of a point in $\Delta$, the fibration $\mathcal{F}|_{U}$ is topologically conjugate to a nodal fibration (see Example 2.6);

2. 2.

   for $n=3$, there is a finite covering $\{U_{i}\}$ of $\Delta$ with open subsets of $B$ such that one of the following three possibilities occur (see also Figure 1):

   1. (a)

      the pair $(U_{i},U_{i}\cap\Delta)$ is homeomorphic to $(D^{3},D^{3}\cap\mathscr{C}_{d})$ and $\mathcal{F}|_{U_{i}}$ is topologically conjugate to either a positive or a negative fibration (see Examples 2.10 and 2.8);

   2. (b)

      the pair $(U_{i},U_{i}\cap\Delta)$ is homeomorphic to $(D^{3},D^{3}\cap\mathscr{C}_{a})$ and $\mathcal{F}|_{U_{i}}$ is topologically conjugate to an alternative negative fibration (see Example 2.9);

   3. (c)

      the pair $(U_{i},U_{i}\cap\Delta)$ is homeomorphic to $(D^{3},D^{3}\cap\mathscr{C}_{e})$ and $\mathcal{F}|_{U_{i}}$ is topologically conjugate to a generic-singular fibration (see Example 2.7);

   We denote by $\Delta_{d}$ the set of points in $\Delta$ belonging to a $U_{i}$ satisfying $(a)$, which are the vertices of $U_{i}\cap\Delta$. We call these points vertices of $\Delta$. We denote by $\Delta_{a}$ the union of the sets $U_{i}\cap\Delta$, where $U_{i}$ satisfies $(b)$; we can assume these sets to be pairwise disjoint. A point in $\Delta$ admitting open neighborhood $U$ of $B$ such that $(U,U\cap\Delta)$ is homeomorphic to $(D^{3},D^{3}\cap\mathscr{C}_{e})$ is called an edge point. We denote by $\Delta_{g}$ the set of edge points.