On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials

We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order formula defining the family. As an important consequence, we derive that the number of distinct homotopy types of semi-algebraic subsets of ${\mathbb{R}}^k$ defined by a quantifier-free first order formula $\mathrm{\Phi }$, where the sum of the additive complexities of the polynomials appearing in $\mathrm{\Phi }$ is at most $a$, is bounded by $2^{(k+a{)}^{O(1)}}$. This proves a conjecture made by Basu and Vorobjov [On the number of homotopy types of fibres of a definable map, J. Lond. Math. Soc. (2) 2007, 757–776].