Expansions of the real line by open sets: o-minimality and open cores
Volume 162 / 1999
Fundamenta Mathematicae 162 (1999), 193-208
DOI: 10.4064/fm-162-3-193-208
Abstract
The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is o-minimal.
