Topology of definable Hausdorff limits

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Let A ⁿ⁺ʳ be a set definable in an o-minimal expansion § of the real field, A' ʳ be its projection, and assume that the non-empty fibers Aₐ ⁿ are compact for all a ∈ A' and uniformly bounded, 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 Aₐᵢ, we give an upper-bound for the Betti numbers bₖ(L) in terms of definable sets explicitly constructed from a fiber Aₐ. 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 relative closure to construct the o-minimal structure § 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.