Let AβRn+r be a set definable in an o-minimal expansion Β§ of the
real field, Aβ²βRr be its projection, and assume that the non-empty
fibers AaββRn are compact for all aβAβ² and uniformly bounded,
{\em i.e.} all fibers are contained in a ball of fixed radius B(0,R). If L
is the Hausdorff limit of a sequence of fibers Aaiββ, we give an
upper-bound for the Betti numbers bkβ(L) in terms of definable sets
explicitly constructed from a fiber Aaβ. In particular, this allows to
establish effective complexity bounds in the semialgebraic case and in the
Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative
closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian
functions in a way that is adapted to complexity problems. Our results can be
used to estimate the Betti numbers of a relative closure (X,Y)0β in the
special case where Y is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and
notations in an attempt to be clearer, references adde