  The aim of this paper is to generalize and improve two of the main
model-theoretic results of "Stable group theory and approximate subgroups" by
E. Hrushovski to the context of piecewise hyperdefinable sets. The first one is
the existence of Lie models. The second one is the stabilizer theorem. In the
process, a systematic study of the structure of piecewise hyperdefinable sets
is developed. In particular, we show the most significant properties of their
logic topologies.
