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

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