Population-Based Structural Health Monitoring (PBSHM), aims to leverage information across populations of structures in order to enhance diagnostics on those with sparse data. The discipline of transfer learning provides the mechanism for this capability. One recent paper in PBSHM proposed a geometrical view in which the structures were represented as graphs in a metric "base space" with their data captured in the "total space" of a vector bundle above the graph space. This view was more suggestive than mathematically rigorous, although it did allow certain useful arguments. One bar to more rigorous analysis was the absence of a meaningful topology on the graph space, and thus no useful notion of continuity. The current paper aims to address this problem, by moving to parametric families of structures in the base space, essentially changing points in the graph space to open balls. This allows the definition of open sets in the fibre space and thus allows continuous variation between fibres. The new ideas motivate a new geometrical mechanism for transfer learning in data are transported from one fibre to an adjacent one; i.e., from one structure to another.