Topology of Definable Hausdorff Limits

Let A ⊆ R be a set definable in an o-minimal expansion S of the real field, A′ ⊆ R be its projection, and assume that the non-empty fibers Aa ⊆ R n are compact for all a ∈ A′ and uniformly bounded. 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 generic fiber Aa. In particular, this allows to establish explicit complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, the case where p = 1 is a special case of the relative closure introduced by Gabrielov. Introduction Let us consider a bounded subset A ⊆ R which is definable in an o-minimal expansion S of the real field (the reader can refer to [5] or [7] for definitions). Let A be the canonical projection of A in R, and for all a ∈ A, we define the fiber Aa as Aa = {x ∈ R n | (x, a) ∈ A}. Assume that these fibers are compact for all a ∈ A. Note that since we assumed that A was bounded, the fibers Aa are all contained in a ball B(0, R) for some R > 0. Recall that for compact subsets A and B of R, we can define the Hausdorff distance between A and B as dH(A,B) = max x∈A min y∈B |x− y|+max y∈B min x∈A |x− y|. The Hausdorff distance gives the space Kn of compact subsets of R n a metric space structure. If (ai) is a sequence in A , and L is a compact subset of R such that the limit of the sequence dH(Aai , L) is zero, we call L the Hausdorff limit of the sequence Aai . It is a well-established fact that the Hausdorff limit L is definable in S : it was first proved by Bröcker [2] in the algebraic case; in the general case, it follows from the definability of types [19, 21]. Recently, direct proofs were suggested in [8] and [18]. In this paper, we investigate how the topology of the Hausdorff limit can be related to the topology of the fibers Aa and its Cartesian powers. To do so, we need to introduce for any integer p a distance function ρp on (p+ 1)-tuples (x0, . . . ,xp) of points in R n by ρp(x0, . . . ,xp) = ∑ 0≤i 0, where the set D a(δ) is the expanded diagonal D a(δ) = {(x0, . . . ,xp) ∈ (Aa) p+1 | ρp(x0, . . . ,xp) ≤ δ}.