On the Reeb Spaces of Definable Maps

We prove that the Reeb space of a proper definable map $$f:X \rightarrow Y$$ in an arbitrary o-minimal expansion real closed field is realizable as quotient. This result can be seen analog Stein factorization morphisms algebraic geometry. also show Betti numbers f arbitrarily large compared to those X, unlike special case graphs manifolds. Nevertheless, when semi-algebraic and X bounded, we singly exponential upper bound on terms number degrees polynomials defining Y, f.