We introduce a notion of interpolation from one topology sigma to another tau assuming that sigma is a subset of tau. Such interpolations are certain transfinite sequence of topologies. We consider the question of whether an interpolation is successful, that is, whether it reaches tau. Answers to this question involve descriptive set theoretic conditions on the relationship between sigma and tau. This theme arose in investigations concerning canonical upper approximations of Borel equivalence relations, but the talk will be independent of these considerations.