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Abstract

We prove that the boolean algebras of sets definable in elementarily equivalent o-minimal expansions of real closed fields are back-and-forth equivalent, and in particular elementarily equivalent, in the language of boolean algebras with new predicates indicating the dimension, Euler characteristic and open sets. We also show that the boolean algebra of semilinear subsets of $[0,1]^n$ definable in an o-minimal expansion of a real closed field is back-and-forth equivalent to the boolean algebra of definable subsets of $[0,1]^n$ definable in the same o-minimal expansion, in the language of boolean algebras with new predicates indicating the dimension, Euler characteristic and open sets, as well as related results.