We introduce a notion of filtration from one topology $\sigma$ to another $\tau$ assuming that $\tau$ contains $\sigma$. Such filtrations are certain transfinite sequence of topologies interpolating between $\sigma$ and $\tau$. We consider the question of whether a filtration succeeds in reaching $\tau$, and, if it does, at what stage it happens. Answers to these questions involve descriptive set theoretic conditions on the relationship between $\sigma$ and $\tau$. This theme arose in investigations concerning the Scott analysis of certain definable equivalence relations, but the talk will be independent of these considerations.