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