Elementary equivalence of lattices of open sets definable in o-minimal expansions of real closed fields

It is well-known that any two real closed fields R and S are elementarily equivalent. We can then consider some simple constructions of new structures out of real closed fields, and try to determine if these constructions, when applied to R and S, give elementarily equivalent structures. We can for instance consider def(R, R), the ring of definable functions from R to R, and we obtain without difficulty that the rings def(R, R) and def(S, S) are elementarily equivalent ([A]). However, if we consider cdef(R, R), the ring of continuous definable functions from R to R, the situation becomes more complicated: Unpublished results of M. Tressl show that, for n > 1, cdef(R, R) defines the set of constant functions with integer value, by a formula that is independent of R and n. Therefore we may have cdef(R, R) 6≡ cdef(S, S), for instance if one field is Archimedean and the other not. It shows that introducing conditions linked to the topology of the real closed field may present an obstacle to elementary equivalence. To understand the situation better it is natural to consider simpler structures than rings of continuous definable functions, but that still demand some topological information from the field. This is what we do in this paper, where we consider the lattices of open definable sets. We show in particular, in Corollary 2.16, that if R and S are elementarily equivalent o-minimal expansions of real closed fields, then the lattices of open definable subsets of R and of open definable subsets of S are L∞ω-elementarily equivalent in the language of bounded lattices expanded by predicates for the dimension and Euler characteristic. The proof is done by a back-and-forth argument. It is worth noting that by [G, Corollary 1] and for n > 1, the lattice of semi-algebraic open subsets of R (for R real closed field) is undecidable. In