  We study the topological complexity of sets defined using Khovanskii's
Pfaffian functions, in terms of an appropriate notion of format for those sets.
We consider semi- and sub-Pfaffian sets, but more generally any definable set
in the o-minimal structure generated by the Pfaffian functions, using the
construction of that structure via Gabrielov's notion of limit sets. All the
results revolve around giving effective upper-bounds on the Betti numbers (for
the singular homology) of those sets.
  Keywords: Pfaffian functions, fewnomials, o-minimal structures, tame
topology, spectral sequences, Morse theory.
