  Let ${\mathcal S}(\R)$ be an o-minimal structure over $\R$, $T \subset
\R^{k_1+k_2+\ell}$ a closed definable set, and $$ \displaylines{\pi_1:
\R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, \pi_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \
\pi_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps.
  For any collection ${\mathcal A} = \{A_1,...,A_n\}$ of subsets of
$\R^{k_1+k_2}$, and $\z \in \R^{k_2}$, let $\A_\z$ denote the collection of
subsets of $\R^{k_1}$, $\{A_{1,\z},..., A_{n,\z}\}$, where $A_{i,\z} = A_i \cap
\pi_3^{-1}(\z), 1 \leq i \leq n$. We prove that there exists a constant $C =
C(T) > 0,$ such that for any family ${\mathcal A} = \{A_1,...,A_n\}$ of
definable sets, where each $A_i = \pi_1(T \cap \pi_2^{-1}(\y_i))$, for some
$\y_i \in \R^{\ell}$, the number of distinct stable homotopy types of $\A_\z,
\z \in \R^{k_2}$, is bounded by $ \displaystyle{C \cdot n^{(k_1+1)k_2},} $
while the number of distinct homotopy types is bounded by $ \displaystyle{C
\cdot n^{(k_1+3)k_2}.} $ This generalizes to the general o-minimal setting,
bounds of the same type proved in \cite{BV} for semi-algebraic and
semi-Pfaffian families. One main technical tool used in the proof of the above
results, is a topological comparison theorem which might be of independent
interest in the study of arrangements.
