Constructible structures and constructible open cores

Abstract

We give a sufficient condition for definably complete expansions \({\mathcal {R}}\) of ordered groups to have constructible open cores. We also prove that the dimension function satisfies the axioms proposed by van den Dries if \({\mathcal {R}}\) is Baire and every set definable in \({\mathcal {R}}\) is constructible. Using the first result, we give necessary and sufficient conditions for definably complete ordered groups (or definably complete ordered fields) to have locally o-minimal (or d-minimal) open cores. As a corollary, we obtain that, if the theory of \({\mathcal {R}}\) is strong, \({\mathcal {R}}\) has a locally o-minimal open core.