Topology of definable Hausdorff limits

Let $A\subset {\mathbb{R}}^{n+r}$ be a set definable in an o-minimal expansion $\mathrm{\S }$ of the real field, $A^{\mathrm{\prime }}\subset {\mathbb{R}}^r$ be its projection, and assume that the non-empty fibers $A_a\subset {\mathbb{R}}^n$ are compact for all $a\in A^{\mathrm{\prime }}$ and uniformly bounded, {\em i.e.} all fibers are contained in a ball of fixed radius $B(0,R)\mathrm{.}$ 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 $A_a\mathrm{.}$ 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 ${\mathrm{\S }}_{\text{pfaff}}$ 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.