  In finite group theory, chief factors play an important and well-understood
role in the structure theory. We here develop a theory of chief factors for
Polish groups. In the development of this theory, we prove a version of the
Schreier refinement theorem. We also prove a trichotomy for the structure of
topologically characteristically simple Polish groups.
  The development of the theory of chief factors requires two independently
interesting lines of study. First we consider injective, continuous
homomorphisms with dense normal image. We show such maps admit a canonical
factorization via a semidirect product, and as a consequence, these maps
preserve topological simplicity up to abelian error. We then define two
generalizations of direct products and use these to isolate a notion of
semisimplicity for Polish groups.
