  In these notes we give a brief introduction to decomposition theory and we
summarize some classical and well-known results. The main question is that if a
partitioning of a topological space (in other words a decomposition) is given,
then what is the topology of the quotient space. The main result is that an
upper semi-continuous decomposition yields a homeomorphic decomposition space
if the decomposition is shrinkable (i.e. there exist self-homeomorphisms of the
space which shrink the partitions into arbitrarily small sets in a controllable
way). This is called Bing shrinkability criterion. It is applied in major
4-dimensional results: in the disk embedding theorem and in the proof of the
4-dimensional topological Poincare conjecture. It is extensively applied in
constructing approximations of manifold embeddings in dimension at least 5, see
for example Edwards's cell-like approximation theorem. We prove for example
that every cell-like subset in a 2-dimensional manifold is cellular, that
Antoine's necklace is a wild Cantor set, that in a complete metric space a usc
decomposition is shrinkable if and only if the decomposition map is a
near-homeomorphism and that every manifold has collared boundary.
