Topological data analysis has been used to understand the structure of the descriptor space of molecules from the point of view of solubility. We have created a mapper visualisation of the descriptor space that revealed that the presence of chlorine is a much greater detriment to solubility in systems with 2 rings than in other cases. Furthermore, the `chemical intuition' type descriptors (molecular weight, ring count) are prominent features in this space, reflecting their importance to chemical properties. Furthermore, a representation of the chemical space was generated using persistent homology applied to molecular graphs. Links between this chemical space and the descriptor space were shown to be in agreement with chemical heuristics. Lastly, norms on persistence landscapes allow the conversion of discrete shape descriptors (atom count, ring count) to continuous ones, and a perspective of the application of these descriptors to QSPR problems is presented.