In set theory, a branch of mathematics, the difference hierarchy over a pointclass is a hierarchy of larger pointclasses generated by taking differences of sets. If àis a pointclass, then the set of differences in àis . In usual notation, this set is denoted by 2-ÃÂ. The next level of the hierarchy is denoted by 3-àand consists of differences of three sets: . This definition can be extended recursively into the transfinite to ñ-àfor some ordinal ñ.
In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the difference hierarchy over à<sup>0</sup><sub style="margin-left:-0.6em">ó</sub> give ÃÂ<sup>0</sup><sub style="margin-left:-0.6em">ó+1</sub>.