Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of
the o-minimal structures, aimed at capturing some finer features of structures
arising from algebraic geometry and Hodge theory. Sharp o-minimality associates
to each definable set a pair of integers known as \emph{format} and
\emph{degree}, similar to the ambient dimension and degree in the algebraic
case; gives bounds on the growth of these quantities under the logical
operations; and allows one to control the geometric complexity of a set in
terms of its format and degree. These axioms have significant implications on
arithmetic properties of definable sets -- for example, \so-minimality was
recently used by the authors to settle Wilkie's conjecture on rational points
in Rexp-definable sets.
In this paper we develop some basic theory of sharply o-minimal structures.
We introduce the notions of reduction and equivalence on the class of
\so-minimal structures. We give three variants of the definition of
\so-minimality, of increasing strength, and show that they all agree up to
reduction. We also consider the problem of ``sharp cell decomposition'', i.e.
cell decomposition with good control on the number of the cells and their
formats and degrees. We show that every \so-minimal structure can be reduced to
one admitting sharp cell decomposition, and use this to prove bounds on the
Betti numbers of definable sets in terms of format and degree