  Let $A\sub \R^{n+r}$ be a set definable in an o-minimal expansion $\S$ of the
real field, $A' \sub \R^r$ be its projection, and assume that the non-empty
fibers $A_a \sub \R^n$ are compact for all $a \in A'$ 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 $A_{a_i},$ we give an
upper-bound for the Betti numbers $b_k(L)$ in terms of definable sets
explicitly constructed from a fiber $A_a.$ 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 $(X,Y)_0$ in the
special case where $Y$ is empty.
