We wish to amplify the idea that the smooth topology of the $S^{1}$-fibration map is subtle. Given a $T^{2}$-fibration $M\to\mathcal{B}$ say, the $T^{2}$-invariant smooth functions on $M$ descend into a sheaf of functions on the base, sitting between the sheaf of smooth functions on $\mathcal{B}$ and the sheaf of continuous functions on $\mathcal{B}$. An example of such a function on our model space is $\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}$. Had we chosen a different $a_{22}$ to begin with, this sheaf would be different. This means assigning a smooth topology on the compactification of a torus bundle across the discriminant locus, is a problem which involves extra data. In general this sheaf depends on functions along $\mathfrak{D}_{i}$, so carries an infinite amount of information, and is therefore expected to be unstable under deformation. This subtlety is related to Joyce’s observation that special Lagrangian fibrations can fail to be given by smooth maps (cf. review Section 1.1.5 and Section 4.12).