This example is the topological model for the fibration over a point of $\Delta$ in the case $n=2$. Let $D$ be the unit disc in $\mathbb{C}$ and $D^{\ast}=D-\{0\}$. Let $f_{0}:X_{0}\rightarrow D^{\ast}$ be a $T^{2}$-bundle with monodromy generated by $\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right)$. We can use Proposition 2.4 to compactify $X_{0}$ as follows. The monodromy invariant cycle, $L\in H_{1}(f_{0}^{-1}(b),\mathbb{Z})$, induces a fibre preserving $T(L)$ action, with $T(L)=L\otimes\mathbb{R}/\penalty L$. The quotient modulo this action yields an $S^{1}$-bundle $\pi_{0}:X_{0}\rightarrow Y_{0}=D^{\ast}\times S^{1}$. One can verify that $\pi_{0}$ extends to an $S^{1}$-bundle $\pi^{\prime}:X^{\prime}\rightarrow Y^{\prime}=D\times S^{1}-\{(0,p)\}$, where $p\in S^{1}$. Furthermore $c_{1}(\pi^{\prime})=\pm 1$. Then Proposition 2.4 ensures that $X^{\prime}$ compactifies to a manifold $X=X^{\prime}\cup\{pt\}$ and that there is a proper map $\pi:X\rightarrow Y=D\times S^{1}$ extending $\pi^{\prime}$. Defining $P:Y\rightarrow D$ as the projection map, we obtain a fibration $f=P\circ\pi:X\rightarrow D$ extending $f_{0}$. The only singular fibre, $f^{-1}(0)$, is homeomorphic to $T^{2}=S^{1}\times S^{1}$ after $S^{1}\times\{x\}\subset T^{2}$ is collapsed to $x$. We denote this fibre by $I_{1}$, following Kodaira’s notation for singular fibres of elliptic fibrations. In Hamiltonian mechanics, a Lagrangian fibration with this topology is known as a focus-focus fibration.